Novel Method for Multi-Dimensional Mapping of Higher Order Modulations for BICM-ID Over Rayleigh Fading Channels

1 No v el Method for Multi-Dimensional Mapping of Higher Order Modulations for BICM-ID Ov er Rayleigh Fading Channels Hassan M. Nav azi and M d. Jahangir Hossain, Me mber , IEEE The Uni v ersity of B r itish Columbia, K elo wna, BC, Canada hnavazi@alumni.ubc.ca , jahangir . hossain@ubc.ca Abstract Multi-dimen sional (MD) map ping o ffers more ﬂexibility in map p ing design for bit-interleaved coded modulatio n with iter ativ e decod ing (BICM-ID) and potentially improves the b andwidth efﬁciency . Howe ver , for higher order signal constellations, ﬁnding suitab le MD mappings is a very complicate d task due to the large num ber of po ssible mapp ings. In this paper, a novel mapping meth o d is in troduced to construc t efﬁcient MD mapping s to improve the error p erform ance of BICM-ID over Rayleigh fading ch a nnels. W e prop ose to br eak the MD map ping design prob le m into four d istinct 2 -D mappin g f unctions. The 2 -D ma p pings are designed such that the resulting MD mapp ing improves the BICM-ID error perfo r mance at low signal to noise ratios ( SNRs). W e also dev elop cost functio ns that can b e op timized to improve the error perf ormance at hig h SNRs. The prop osed mapping method is very simple compar ed to well-known mapping method s, and it can achieve suitab le MD mappings for different modulatio ns including higher order mo dulations for BICM-ID. Simu lation r esults sh ow th a t o ur m apping s signiﬁcantly outp e r form the previously known mapping s at a target bit er ror rate ( BER) of 1 0 − 6 . Our mapping s also offer a lower er r or-ﬂoor comp ared to their well-known co unterpa r ts. Index T erms BICM-ID, multi-dim e n sional signal map ping, QAM, Rayleigh fading channels. 2 I . I N T R O D U C T I O N T rellis cod ed modulation ( TCM) [1] i m proved the bit error rate (BER) performance of coded modulation by maximizing the Euclidean distance among the coded s i gnal sequences. T o i mprove the BER of TCM over Rayleigh fading channels, bi t interleav ed coded modul ation (BICM) was introduced by Zehavi [2]. BICM off ers good performance over Rayleigh fading channels. Howe ver , the random modulation caused by th e interleav er degrades the BICM performance over additive white Gaus sian no i se (A WGN) channels. T o address this problem , iterative decoding was used at the receiver . The resulting sy stem is referred to as BICM with iterative decoding (BICM-ID) and is in vestigated in [3]-[5]. BICM-ID offe rs better performance over A WGN and Rayleigh fading channels. BICM can also use oth er iterative decoding schemes such as low density parity check code (LDPC). In [20], it i s demonstrated that BICM-ID with signal space diversity (BICM-ID-SSD) outperforms the LDPC-BICM over fading channels. BICM-ID with o ut SSD can also outperform LDPC-BICM when the number of iterations at the decoder is sm aller than a certain number , which m akes the d ecoder sim pler . Moreover , compared to LDPC-BICM, BICM-ID uses a simple con volutional code instead of a m ore complex LDPC code. As such, BICM-ID offers a lower system complexity . Cons equently , BICM-ID is a go o d candidate as a coded modulation especially when the sys t em com p lexity becomes a more important concern. It is widely known that the signal labeling map (mapping) pl ays a crucial role in BIC M-ID performance[6]. Signal labeli n g is deﬁned as the assignment of a binary sequence to a singl e symbol from a signal constellation. It is also referred to as multi-dim ensional (MD) mapping when a sequ ence of binary bits is mapped t o a vector of symb ols instead of a singl e s ymbol. MD mapping is m ore ﬂexible to desig n and also offers better bandwidth ef ﬁciency [7]. The MD labeling process is more adaptable to d i f ferent design guidelines because MD space provides more diverse Euclidean distances among symbols. In [8], MD labeling was used for TCM and made the syst em’ s BER better through using the a vailable bandwidt h more ef ﬁciently . This dev elopment moti vated researchers to use the MD labeling technique and i ts efﬁcient u s e of bandwidt h to further improv e the BER of BICM-ID [7], [9]-[12]. The MD labeling techni q ue can also be applied to h i gher order modulati on to increase 3 the data rate of BICM-ID because using a lar ger constellation makes i t poss ible to send more bits in the same s i gnaling rate. Ho we ver , providing a suitable M D labeling of a large constellation for BICM-ID is very challenging because of t he lar g e number of possible mappings. In general, for a 2 N -D 2 m -ary m odulation, th ere are 2 mN ! possible mappin g s, where ! denot es the factorial operation. For example, the nu mber of p ossible 4 -D mapping s for a 256 -ary quadrature am p litude modulati o n ( 256 -QAM) is 5 . 16 × 10 287193 , which is an astronomical ﬁgure. It is im prtant to note that MD mapping improves the perforamnce of BICM-ID at a particular expense of the sys tem’ s complexity [10]-[11]. Howe ver , t he study of i ts complexity is beyond the scope of this paper . Diffe rent labeling approaches for BICM-ID such as the genetic algorithm (GA), reactive tabu search (R TS) algorithm, extrinsic information transfer (EXIT)-based search algorithm , bin ary switching algo ri thm (BSA), and random labeling technique h ave been extensiv ely in ve stigated [6]- [25]. Indeed, high computation al compl exity is the main pitfall of all the propo sed computer search based metho ds i n the literature when l o oking for a goo d mapping of a large constell ation. The GA and R TS algorithms hav e been used in [14] and [15], respective ly , to ﬁnd t h e optim um mappings for BICM-ID. In these studies, ho we ver , the authors ha ve not re ported results for constellations larger than 64 -QAM due to a very high computatio n al complexity . In [16]-[19], an EXIT -based method is proposed to ﬁnd su itable mappings that improve the i terativ e decoding systems BER at any sign al to noise ratio (SNR) wi th an arbitrary number of it eratio n s. In [20], the authors hav e designed an EXIT -chart aided serach method to de velop capacity approaching coded modulation s . In particular , they hav e prop osed a BIC M-ID system with signal space di versity that approaches the channel capacity in both the fading and non-fading channels. Howe ver , the EXIT - based search con s iders b o t h the modul ato r and encoder in detail and also th e iterations between the decoder and demodu lator . This makes t he process compli cated for ﬁnding a good mapping of a lar ge M D constellation amo n g a large n umber of possi ble labelings. The binary switchi n g algorithm (BSA) [21], which is the best known m apping search method for BICM-ID, becomes intractable when searching for a good mapping of an MD constellation due to the hu ge search complexity [9],[11]. Although it is demonstrated in [9] and [13] that the random l abeling technique results i n 4 suitable mappings for BICM-ID, it is st i ll com p utationally complex to look for a good mapping of a lar ge constellati on. This is because the random labeling m ethod s earches for a good mappi n g among a lar ge set of randomly genera ted mappin g s. In addi t ion t o th ese comput er search based method s, a heuristi c method has also been explored in [12] to const ruct MD labelin g for BICM-ID. But, this method is limited to 16 - and 64 -QAM. Mo reov er , t his method is not d esi gned for Rayleigh fading channels. In this paper , we p ropose a novel m ethod to develop suit able MD mappi ngs to improve the BER performance of BICM-ID over Rayleigh fading chann els in both the low and high SNR regions. T o improve the BER performance of the system , we increase the harmo n ic mean of t h e m inimum squared Euclid ean d istance (MSED) [6],[22] of fered b y the m app i ng. The reported analytical and numerical results conﬁrm the efﬁcienc y of the achie ved m appings. The novelty and contributions of our work are as follows. (i) W e break the M D labeling problem into four d i stinct 2 -D mappi n gs, which makes it easier t o optimi ze t he MD labeling . (ii) W e desig n the four 2 -D mappings such t h at the resulting MD mappi ng improves the BER of BICM-ID at low SNRs. (iii) W e de velop cost functions t hat can be optim ized ove r the 2 -D mappings to improve th e performance of the resulting M D mapping’ s at high SNRs. (iv) W e dev elop ef ﬁcient MD mappings of higher order m odulations includ i ng 2 m -ary ( m = 4 , 5 , ..., 10 ) mo d ulations. Finally , (v) we propose ef ﬁcient MD mappings of di fferent modulat i on types including QAM, phase shift keying (PSK), and irregular modulations. Compared to the labeling meth o d in [12], the proposed metho d in thi s paper has the following additional contributions. (i) The presemnt stu d y is not lim ited to a particular modulation type, i.e., it is not modulatio n spesi ﬁc. Ho we ver , the method in [12] is on ly for square QAMs. (i i) The proposed m ethod in this paper can generate efﬁcient mappi ngs for m odulations with diffe rent orders while the m ethod in [12] is only for 16 - and 64 -ary parti cular mo d ulations. (iii) Finally , the proposed method in this paper is designed speciﬁcally for Rayleigh fading channels. As a result, over fading channels, the resulting m app i ngs outperform th e mappings in [12]. The rest of this paper is organized as follows. The BICM-ID system model is d escrib ed in Section II. The proposed MD labeli ng method is i ntroduced in Section III. In Section IV, cost functions 5 Π Π 1 − Π y x ɶ u ( ) e L ɶ v c u E n c o d e r D ec o d e r M o d u l a t o r D e m o du la t o r C ha n ne l v ( ) a L ɶ v Fig. 1. The block diagram of a BICM-ID system. are de veloped and optimi zed to obtain good MD mappings. Numerical result s and discussions are presented in Section V. Finally , Section VI summarizes the conclusion s. I I . S Y S T E M M O D E L The block diagram of a con ventional BICM-ID syst em is shown in Fig. 1. The transm i tter is constructed from serial concatenation of a con volutional encoder , a bit interleav er and a modulator . In the system considered in this paper , the modulator maps a sequence of mN bi ts to a vector of N consecutiv e 2 m -ary si gnal points, using a MD m apping function µ : { 0 , 1 } mN − → χ = χ N . Let x ∈ χ be a 2 N -D signal poin t represented as x = ( x 1 , x 2 , · · · , x N ) , (1) where x i ∈ χ . x i s labeled by an mN -bit binary sequence v as x = µ ( v ) . (2) The average ener gy per signal -vector is assumed to be 1 , i .e., E x = 1 . It is assumed that the channel state informatio n is kno wn at the receiver . The received signal-vector corresponding to t he 6 transmitted signal-vector x is gi ven by y = h x + n , (3) where h is t he Rayleigh fa ding coef ﬁcient corresponding to x , and n is a vector of N addi tiv e complex white Gaussi an nois e samples with zero-mean and variance N 0 . W e assume t hat the fading coef ﬁcient for all signals in a signal-vector remains the sam e. Clearly , for the A WGN channels h = 1 . At the recei ver , from the recei ved signal y t and t h e a priori log-li kelihood ratio (LLR) of th e coded bit s, the demapper computes the e xtrinsic LLR for each of the b its in the receiv ed sym bol as described i n [4]. Then, the extrinsic LLRs are permuted by t he random deinterlea ver and used by the channel decoder . The decoder then calculates the extrinsic LLR o n the coded bits using the BCJR algorithm [26]. T h ese LLRs are int erlea ved and then fed back to the demapper to be used as the a pr iori LLRs in the next iteratio n. I I I . P RO P O S E D M A P P I N G M E T H O D As mentioned earlier , for a 2 N -D 2 m -ary mo d ulation there are 2 mN ! possib le mappings. In fact, a comprehensiv e computer search to ﬁnd good mappings becomes intractable quickly as the modulation order increases. Even the well-known BSA mapping s earch method cannot be used directly to obtai n good MD m appings of high er order modulatio n s. T h erefore, we propose an ef ﬁcient technique to ﬁnd good MD mappings for BICM-ID systems over Rayleigh fading channels. A. Mapping Design Guid el i ne The performance of BICM-ID ove r Rayleigh fading channels is inﬂuenced by the harmonic mean of the MSED, which is calculated for a g iven m app i ng function, µ , applied to signal set χ . For a 2 N -D mappin g of a 2 m -ary constellati on, th e harmoni c m ean of t he MSED is gi ven by [10] ,[22] Φ( µ, χ ) =   1 mN 2 mN mN X i =1 1 X b =0 X x ∈ χ i b 1 k x − ˆ x k 2   − 1 , (4) 7 where x = ( x 1 , x 2 , · · · , x N ) is a 2 N -D signal poi n t and χ i b is the subset of s i gnal point s in χ whose labels take value b at the i th bit position. For the perf ormance in the l ow SNR region, ˆ x = ( ˆ x 1 , ˆ x 2 , · · · , ˆ x N ) refers t o the n earest neig h b or 1 of x i n χ i ¯ b , and (4) is referred t o as the harmonic mean of the MSED before feedback. For the asympto tic performance (performance in the high SNR re gion), χ i ¯ b in v olves only one symbol-vector ˆ x , which is dif ferent from x only in the i th bit position [22]. In t his case, (4) is referred to as the harmonic m ean of t h e MSED after feedback, which is denot ed by ˆ Φ( µ, χ ) . T o achiev e good p erformance in the l ow and high SNR regions, a large va lue of Φ( µ, χ ) and ˆ Φ( µ, χ ) is required, respectively . Howe ver , maximizing (4) i s a very comp l ex problem ev en for a modulation such as 6 4 -QAM [15]. W e prop o s e an inn ovati ve approach to generate MD m appings using 2 -D m appings such that the result ing MD m apping has a g reater value of Φ( µ, χ ) . Later , we dev elop cost functions t h at are o ptimized over the employed 2 -D mappings to achiev e a hig h v alue of ˆ Φ( µ, χ ) for the MD mapping. Our cost functions are very simple and give excellent results, e ve n for hig her order constellati ons s uch as MD 1024 -QAM. B. MD Mapping Usi ng 2 -D Map pings Let l = ( l 1 , l 2 , · · · , l mN ) be an mN -bit bin ary label. W e can write l = ( l 1 , l 2 , · · · , l N ) , where l i is a m -bit binary label and is g iv en by l i = ( l ( i − 1) m +1 , · · · , l im ); i = 1 , ..., N . (5) Suppose that L denotes the set of all mN -bit binary labels and L e and L o represent t he subset of all l ∈ L with e ven and odd Hamming w eig hts, respectiv ely . The M D mapping problem can be broken into four mappin gs in 2-D s ignal space as described below . According to the proposed MD mapping functi on, i.e., µ , label l is mapped to the 2 N -D sig nal point x = ( x 1 , · · · , x N ) as given below 1 Th symbol-v ectors with the mi nimum Euclidean distance are refereed to as the nearest neighbo urs. 8 x i =                    λ el ( l i ) if i = 1 , l ∈ L e , λ ol ( l i ) if i = 1 , l ∈ L o , λ er ( l i ) if i > 2 , l ∈ L e , λ or ( l i ) if i > 2 , l ∈ L o , ; i = 1 , ..., N , (6) where λ el , λ ol , λ er , and λ or are 2 -D mappi ng functions, whi ch wil l be discussed later in this section. In the applied mapping, l et χ e and χ o represent the subset of signal points i n χ whose labels belong to L e and L o , respecti vely . W ith out loss of generality , assum e th at x ∈ χ e and ˆ x ∈ χ o where ˆ x = ( ˆ x 1 , ˆ x 2 , · · · , ˆ x N ) is a signal p oint whose label is di f ferent from that of x only in one bit position. W e partit ion the 2 -D signal constellation χ into two s eparate subsets with equal cardin ali ties and denote t h em as χ el and χ ol . Then, we limit the ﬁrst element in x and ˆ x , i.e., x 1 and ˆ x 1 , to b elo ng to χ el and χ ol , respecti vely . In (6), λ el ( . ) and λ ol ( . ) each map an m -bit label to a 2 -D si g nal point chosen from χ el and χ ol , respecti vely . Howe ver , χ el and χ ol in v olve onl y 2 m − 1 signal points while there are 2 m distinct m -bit labels. As a con s equence, each sig nal point in χ el and χ ol should be mapped by two m -bit labels simul taneously . In o rd er t o obtain a one-to-one MD mappin g function, we restrict the two labels th at are mapped to a particular signal point in either χ el or χ ol to be diffe rent i n an odd number of bits . Speciﬁcally , we ass u me they are d iff erent onl y in the ﬁrst bit position. This prohibi ts t he labels wi th a Hamming di stance larger than ( m + 1 ) bits to be mapped to t he nearest neighb o urs in either of χ e or χ o . As a result, the value of Φ( µ, χ ) increases. On the other hand, t here is no constraint on λ er ( . ) and λ or ( . ) except th ey need to be bijective. Pr oposition 1 . Let us assume that l 1 = ( l 1 1 , l 2 1 , · · · , l n 1 ) and l 2 = ( l 1 2 , l 2 2 , · · · , l n 2 ) are two n -bit labels, D = d H ( l 1 , l 2 ) is the Hamming distance between l 1 and l 2 , and W = w 1 + w 2 , wher e w j is t he Hamming weight of l j ( j = 1 , 2 ). If W ∈ E , D ∈ E , and if W ∈ E , D ∈ E . Pr oof. See Appendix A. Pr oposition 2. In the proposed MD mappi n g function , µ , ther e is a one-to-one corr espondence 9 between MD s ignal point s and binary l abels. Pr oof. See Appendix B. Pr oposition 3. In the pr oposed MD mapping f unction, the Hamming distance between the n ear est neighbours in either of χ e or χ o cannot be lar ger than ( m + 1 ) bi ts. Pr oof. See Appendix C. I V . D E V E L O P M E N T A N D O P T I M I Z A T I O N O F C O S T F U N C T I O N S T O I M P R OV E ˆ Φ As mentio ned in Section III-A, maximi zing th e harmonic mean is a very com plicated task. In this section, we use the 2 -D mappings i n (6), i.e., λ el , λ ol , λ er , λ or , to de ve lop t wo cost function s and deri ve a lower bound for ˆ Φ( µ, χ ) , i.e., ∆ . Then, we propose an algorithm to maximize ∆ by optimizing the cost functio n s ov er the 2 -D mapping s . As s u ch, the achie ved 2 -D mappings construct a MD m app i ng that provides a lar ger value of ∆ , which results in a larger value of ˆ Φ( µ, χ ) . A. Development of Cost Functions In order t o achieve a lowe r bound for ˆ Φ( µ, χ ) , ∆ , we de velop an upper b o und for ˆ Φ( µ, χ ) − 1 . In particular , we ﬁrst decompose ˆ Φ − 1 into two equal parts, i.e., Ω e ( µ, χ ) and Ω o ( µ, χ ) , wh ere in Ω e , x ∈ χ e and ˆ x ∈ χ o . Next, we derive an upper bound for Ω e ( µ, χ ) , i.e., Ψ( µ, χ ) , and then, we decompose Ψ in t o Ψ l ( µ, χ ) and Ψ r ( µ, χ ) , wh ere Ψ l uses o n ly the ﬁrst symbol in x and ˆ x , i .e., x 1 and ˆ x 1 , and Ψ l uses th e rest of symbo l s in x and ˆ x . As x ∈ χ e and ˆ x ∈ χ o , x 1 and ˆ x 1 in Ψ l are obtained using the mapping functions λ el and λ ol , respectively , while x i and ˆ x i ( 2 6 i 6 N ) in Ψ r are obtained usin g λ er and λ or , respectiv ely . Thus, we de velop two cost functions ψ l and ψ r , where ψ l generates t he same values as Ψ l by consi dering all the cases o f using λ el and λ ol in Ψ l , and ψ r generates the s ame vales as Ψ r by cons idering all the cases of using λ er and λ or in Ψ r . Finally , we use ψ l and ψ r to dev elop ∆ . In what follows, we dis cuss these s t eps in more detail. 10 W e use (4) to write (7) ˆ Φ( µ, χ ) − 1 = Ω e ( µ, χ ) + Ω o ( µ, χ ) , where (8) Ω e ( µ, χ ) = 1 mN 2 mN mN X i =1 1 X b =0 X x ∈ χ i b x ∈ χ e 1 k x − ˆ x k 2 and (9) Ω o ( µ, χ ) = 1 mN 2 mN mN X i =1 1 X b =0 X x ∈ χ i b x ∈ χ o 1 k x − ˆ x k 2 . The sets χ e and χ o hav e the same cardinality . M o reov er , when a given x is in χ e then the corresponding ˆ x belongs to χ o and vice ver sa. Therefore, (8) and (9) use the same set of Euclidean distances, which result s in Ω e ( µ, χ ) = Ω o ( µ, χ ) . As a result , using (7) we have ˆ Φ( µ, χ ) − 1 = 2Ω e ( µ, χ ) . (10) Since k x − ˆ x k 2 = P N j =1 | x j − ˆ x j | 2 , then (8) can be rewritten as (11) Ω e ( µ, χ ) = 1 mN 2 mN mN X i =1 1 X b =0 X x ∈ χ i b x ∈ χ e 1 P N j =1 | x j − ˆ x j | 2 . Pr oposition 4. Let y = ( y 1 , y 2 , · · · , y N ) be a vector of non-ne gative r eal numbers. Then, we have 1 P N i =1 y i 6 1 N N X j =1 1 y j . (12) Pr oof. See Appendix D. Applying (12) i n (11), we can wri t e Ω e ( µ, χ ) 6 K Ψ( µ, χ ) (13) 11 where K = 1 mN 2 2 mN is a const ant value and Ψ( µ, χ ) is deﬁned as Ψ( µ, χ ) = mN X i =1 1 X b =0 X x ∈ χ i b x ∈ χ e N X j =1 1 | x j − ˆ x j | 2 . W e can decom pose Ψ( µ, χ ) as Ψ( µ, χ ) = Ψ l ( µ, χ ) + Ψ r ( µ, χ ) , (14) where Ψ l ( µ, χ ) and Ψ r ( µ, χ ) are gi ven by Ψ l ( µ, χ ) = mN X i =1 1 X b =0 X x ∈ χ i b x ∈ χ e 1 | x 1 − ˆ x 1 | 2 (15) and Ψ r ( µ, χ ) = mN X i =1 1 X b =0 X x ∈ χ i b x ∈ χ e N X j =2 1 | x j − ˆ x j | 2 . (16) Let l = ( l 1 , l 2 , · · · , l mN ) and ˆ l = ( ˆ l 1 , ˆ l 2 , · · · , ˆ l mN ) are two mN -bit labels, which are diff erent only in the i th bit posi tion, and are m apped to x = ( x 1 , · · · , x N ) and ˆ x = ( ˆ x 1 , · · · , ˆ x N ) , respective ly . W e deﬁne l i and ˜ l i respectiv ely as the i th m -tuple bits of l and ˆ l , and rewrite l = ( l 1 , l 2 , · · · , l mN ) and ˜ l = ( ˜ l 1 , ˜ l 2 , · · · , ˜ l mN ) . Then, Ψ l in (15) i s equal to Ψ ′ l , which i s given by Ψ ′ l ( λ el , λ ol , L ) = mN X i =1 1 X b =0 X l ∈L i b l ∈L e 1 | λ el ( l 1 ) − λ ol ( ˆ l 1 ) | 2 , (17) where L i b ∈ L is the s ubset of labels wi th value b in their i th bit p o sition. For a given m -bit sequence l i , ˆ l i can take ( m + 1) dist inct m -bit sequences, where each one i s the same as l i or differ ent from l i only in one bit positi on. F or e xample, if m = 4 and l i = (0 , 0 , 0 , 0) , ˆ l can take eith er of the 5 labels in { (0 , 0 , 0 , 0) , (0 , 0 , 0 , 1) , (0 , 0 , 1 , 0) , (0 , 1 , 0 , 0) , (1 , 0 , 0 , 0) } . Let α = ( α 1 , · · · , α m ) and β = ( β 1 , · · · , β m ) be t wo bi n ary sequences, where β h as the Hamming distance of either zero or 12 one bit from α . The set of ( m + 1) poss i bilities for β is denoted by B . Assume th at for a given i , l i = α and ˆ l i = β . Then, Ψ ′ l in (17) is equ al to ψ l , which is deﬁned as ψ l ( λ el , λ ol , χ el , χ ) = X α X β ∈B a ( l ) α , β | λ el ( α ) − λ ol ( β ) | 2 , (18) where a ( l ) α , β is com p uted as a ( l ) α , β = mN X i =1 1 X b =0 X l ∈L i b l ∈L e I ( l 1 = α , ˆ l 1 = β ) , (19) where I ( x ) i s an indicator functi on and is deﬁned as I ( x ) =      1 , if x is true , 0 , otherwise . (20) Similarly , Ψ r in (16) i s equal to Ψ ′ r , which i s given by Ψ ′ r ( λ er , λ or , L ) = mN X i =1 1 X b =0 X l ∈L i b l ∈L e N X j =2 1 | λ er ( l j ) − λ or ( ˆ l j ) | 2 . (21) The m -bit elements l i in l = ( l 1 , l 2 , · · · , l N ) are independent from one another for all v alues of i . Then, Ψ ′ r in (21) i s equal to ψ r , which i s deﬁned as ψ r ( λ er , λ or , χ ) = mN X i =1 1 X b =0 X l ∈L i b l ∈L e N − 1 | λ er ( l 2 ) − λ or ( ˆ l 2 ) | 2 , (22) and can be rewritten as ψ r ( λ er , λ or , χ ) = X α X β ∈B ( N − 1) a ( r ) α , β | λ er ( α ) − λ or ( β ) | 2 , (23) 13 where a ( r ) α , β is com p uted as a ( r ) α , β = mN X i =1 1 X b =0 X l ∈L i b l ∈L e I ( l 2 = α , ˆ l 2 = β ) . (24) Using (10), (13), and (14), a lo wer bo u nd of ˆ Φ( µ, χ ) can b e deri ved as foll ows ˆ Φ − 1 ( µ, χ ) 6 2 K (Ψ l ( µ, χ ) + Ψ r ( µ, χ )) (25) ⇒ ∆ 6 ˆ Φ( µ, χ ) , where ∆ is giv en by ∆ = 1 2 K (Ψ l ( µ, χ ) + Ψ r ( µ, χ )) . (26) Because Ψ l ( µ, χ ) = ψ l ( λ el , λ ol , χ el , χ ) and Ψ r ( µ, χ ) = ψ r ( λ er , λ or , χ ) , we rewrite ∆ as ∆ = 1 2 K ( ψ l ( λ el , λ ol , χ el , χ ) + ψ r ( λ er , λ or , χ )) . (27) Note that (27) operates in 2 -D signal space rather than MD signal space, and as a result, opti m ization is much simpler . Our obj ectiv e is to maximize ∆ and then to calculate the corresponding ˆ Φ( µ, χ ) . Since ψ l ( λ el , λ ol , χ el , χ ) and ψ r ( λ er , λ or , χ ) are independent from each other , then the maximum value of ∆ , ∆ max , is g iven by (28) ∆ max = 1 2 K  min λ el ,λ ol ,χ el ψ l ( λ el , λ ol , χ el , χ ) + min λ er ,λ or ψ r ( λ er , λ or , χ )  − 1 . B. Minimizati on of Cost Functions The minim ization of the cost fun ct i ons ψ l and ψ r in (28) can b e done usin g t h e BSA [21]. Not e that the minim ization of ψ r is simpl er than that of ψ l because ψ r deals with fewer effecti ve arguments. Thus, we ﬁrst optimize ψ r , and then, use the obtain ed results to si mplify the m inimization of ψ l . 14 1) Minimization of ψ r : T o minimize ψ r , two random mappings are ini tially considered as λ er and λ or . Then, the BSA is used t o m i nimize ψ r by modifying λ er and λ or . In fact in ψ r , the cost value for a given symbol in λ er is com puted b y using ( m + 1) corresponding symbo ls from λ or . As a result, simultaneously modifyin g λ er and λ or can m ake the optimization complex. Therefore, our approach is to m inimize ψ r by al t ernatingly modifyi n g each of λ er and λ or . In other words, we use the BSA to decrease ψ r by m odify λ er . After a giv en nu m ber of it eratio ns, λ er and λ or are exchanged. Again, th e BSA is used to decrease ψ r by modi fying t he ne w λ er . This procedure is repeated up to a gi ven numb er of iterations. 2) Minimization of ψ l : In additi o n to λ el and λ ol , χ el is another effe ctive argument in comput i ng ψ l . As there is no const rain t on χ el , it is a comp l ex process to minimi ze ψ l . T o simplify t he optimizatio n process, χ el is constrained to in volve o n ly the symbols whose labels in the ob t ained λ er take binary v alue b in a given bit position. In this paper , we assume that χ el in v olves t he symbols whose l abels in λ er take the value zero in the ﬁrst bit-posi t ion. T h e functions ψ l and ψ r are computed by considerin g the simil ar E u clidean distances bet ween two-dimensi o nal symbols. As a result, there is a pot ent ial adv antage in applying the above menti o ned constraint on χ el because it will be easier to ﬁnd a suitable λ el corresponding to a given λ ol . After determin ing χ el and χ ol , two random mapping s are generated as λ el and λ ol . Again the BSA is appl ied to minim i ze ψ l by modifying λ el . Then, λ el is exchanged by λ ol and the BSA minimi zes ψ l by modifying the ne w λ el . This procedure i s repeated u p to a given number of iterations. By exe cuting the propos ed al g orithm for a certain number of iterations , a local maxi mum v alue is calculated using (28). The search algorithm is executed se veral ti mes and each time the corresponding value for ˆ Φ( µ, χ ) is calculated. Finally , the mod ulations corresponding to the maximum obtained ˆ Φ( µ, χ ) are chosen. Fig. 2 illustrates the ﬂowc hart of th e propos ed algorithm . Numerical resul ts conﬁrm that the propo sed algori t hm generates mappings wi t h si gniﬁcantly large values of ˆ Φ( µ, χ ) . As a result, the o b tained mappings would also improve th e error performance of BICM-ID system s in the h igh SNR re gion over Rayleigh fading channels . C. Simpliﬁed BSA 15 W e use the simpliﬁed BSA in our algorithm to reduce the search compl exity . The BSA calculates a cost function for each symbol in an initial rando m m app ing, and t hen, li sts the symbo l s in d escendi ng order i n terms of cost value. Next, the label of the symbol with the highest cost value is switched with the label of another symbol such th at the total cost is reduced as much as possible. After each switch, the BSA again lists the symb ols in descending cost value and repeats the swi t ching process. It is important to note that each switch in the subsequent roun ds of t he BSA af fects t h e cost value of only a li m ited number of sy mbols in the mappi ng. In particular , in maximi zing ˆ Φ( µ, χ ) for a given modulation usi ng the BSA, when the l abel of symbol s i , i.e., l i , is swit ched by label of sy m bol s j , i.e., l j , the cost value changes only for s i and s j and for the sym b ols wh ose labels are differ ent from l i or l j only in one bit position. For example, for a 2 m -ary m odulation, the numb er o f af fected symbols in each switch in a subsequent round is at most ( m + 2 ). Therefore, in the subsequent rounds of the simp l iﬁed BSA, we calcul at e t he cost function only for the s y mbols th at are affec ted by the most recent switch. This modiﬁcation makes the BSA much sim pler while re sulting in the exactly sam e results. V . N U M E R I C A L R E S U L T S A N D D I S C U S S I O N This section provides resulting mappings and selected nu m erical results to illust rate the perfor- mance and adv antage of our prop o sed MD m appings for BICM-ID. A. Resulting MD M a p pings Our proposed algorithm is us ed to obtain MD mappings of diffe rent modulations such as 8 -PSK, the o p timum 8 -QAM [27], and 2 m -QAM for m = 4 , 5 , · · · , 10 . T abl es I-II show the resul ting 2 - D m app ings, i.e., λ er , λ or , λ el , and λ ol , in decimal format for 8 -ary modulati o ns and 16 -QAM . T ABLE I P RO P O S E D λ er , λ or , λ el , A N D λ ol F O R 8 - A RY M O D U L A T I O N S . 8 -PSK Optimum 8 -QAM λ er [2 7 6 5 4 1 0 3] [1 3 5 7 2 0 6 4] λ or [4 1 0 3 2 7 6 5] [6 4 2 0 5 7 1 3] λ el [(2, 6) (1, 5) (0, 4) (3, 7)] [(1, 5) (3, 7) (2, 6) (0, 4)] λ ol [(1, 5) (0, 4) (3, 7) (2, 6)] [(2, 6) (3, 7) (0, 4) (1, 5)] 16 • Choose er λ and or λ randomly • 0 j = Use BSA to minimize r ψ • Exchange er λ with or λ • 1 j j = + . . j it num r < Calculate ∆ max ∆ > ∆ • max ∆ = ∆ • 1 i i = + • Calculate ˆ Φ • Choose el λ and ol λ randomly • 0 k = Use BSA to minimize l ψ • Exchange el λ with ol λ • 1 k k = + . . k it num l < . i it num < End Start Yes No Yes No No Yes Yes No Determine el χ and ol χ max 0 ∆ = , max ˆ 0 Φ = , 0 i = max ˆ ˆ Φ > Φ • max ˆ ˆ Φ = Φ • Save er λ , or λ , el λ , and ol λ No Yes Fig. 2. Flowcha rt of the proposed algorithm ( it.num.r , it.num.l , and it.num represent the number of iterations for dif ferent loops). The results for lar ger mo dulations are repoertd in Apendix E. In these tables, the decimal labels are ordered according to t he symbol o rder in the corre sponding constellations . For 8 -PSK and the optimum 8 -QAM, the considered symbol order i s sh own i n Fig. 3. For square QAMs, it is assumed that the symbol order starts from the top left corner in the cons tellation and increases from top to bottom and from left to right (see Fig. 4 . (a) as an example for 16 -QAM). For cross QAM 17 S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 1 S 2 S 3 S 4 S 5 S 6 S 8 S 7 ( a ) ( b ) Fig. 3. (a). Symbol’ s arrangement in 8 -ary constellations: (a). 8 -PSK and (b). the optimum 8 -QAM [ 27]. 3 ,1 1 ( ) d 2,10 5,1 3 4,12 7,1 5 6,1 4 1 , 9 0, 8 0, 8 4,12 1 , 9 5,13 2,10 6,1 4 7,1 5 3,11 1 2 ( ) c 8 5 4 1 3 9 1 0 1 0 1 1 3 1 4 1 5 2 6 7 3 ( ) b 2 1 5 1 1 7 6 1 4 1 0 0 4 1 2 1 5 8 9 1 3 ( ) a 1 S 6 S 7 S 1 0 S 1 1 S 5 S 2 S 9 S 1 3 S 1 4 S 3 S 4 S 8 S 1 2 S 1 6 S 1 5 S Fig. 4. (a). Symbol’ s arrangement in 16 -QAM, and achie ved 16 -QAM mappings in decimal format: (b). λ er , (c). λ or , and (d). λ el (the unshaded symbols), λ ol (the shaded symbols). constellations, su ch as 32 -QAM, we consider the sym bol order used in [28 ]. In these tables, the resulting 2 -D mappings for higher order modul ations are indicated in multiple rows. For example in T able VII, λ er for 64 -QAM is ind i cated i n two rows, where the ﬁrst element i n the second row is the label of the 33 r d symbol in the 64 -QAM constell ation. In addi t ion, two labels in the i th parentheses in λ el and λ ol in each tabl e belong to the i th symbol i n χ el and χ ol , respectively . For example, Fig. 4.(b), (c), and (d) i l lustrate the 16 -QAM mappi n gs reported in T able II. As menti oned in Section IV - B, χ el in v olves t he s ymbols whose binary labels in λ er take the value zero at the ﬁrst bit p o sition. In other words, χ el is con s tructed by the symbols whose decimal label in λ er is smaller than M 2 . As a result, for 16 -QAM, χ el = { S 1 , S 2 , S 5 , S 6 , S 9 , S 10 , S 13 , S 14 } , where χ el is indicated b y unsh aded symbols in Fig. 4.(d). The remaining 16 -QAM symb o ls belong to χ ol , which are s h aded in Fig. 4.(d). Example 1 clariﬁes h ow t o use λ el , λ ol , λ er , and λ or to construct th e p rop osed MD mappin g of 16 -QAM . Example 1. In t h e propo s ed MD mappi ng metho d , let us set m = 4 ( 16 -QAM), N = 3 and 18 T ABLE II P RO P O S E D λ er , λ or , λ el , A N D λ ol F O R 16 - Q A M . λ er [3 2 15 11 7 6 14 10 0 4 12 13 1 5 8 9] λ or [12 8 5 4 13 9 1 0 10 11 3 7 14 15 2 6] λ el [(3, 11) (2, 10) (7, 15) (6, 14) (0, 8) (4, 12) (1, 9) (5, 13)] λ ol [(5, 13) (4, 12) (1, 9) (0, 8) (3, 11) (7, 15) (2, 10) (6, 14)] l = (0 , 1 , 1 , 0 , 1 , 1 , 1 , 1 , 0 , 1 , 1 , 1) . The label l is considered as a sequence of t hree 4 -bi t labels, i.e., l = ( l 1 , l 2 , l 3 ) , where l 1 = (0 , 1 , 1 , 0) , l 2 = (1 , 1 , 1 , 1) , and l 3 = (0 , 1 , 1 , 1) . The mapping rule in (6) is used t o map l = ( l 1 , l 2 , l 3 ) to sign al point x = ( x 1 , x 2 , x 3 ) , as follows. The Hammi ng weight of l is odd, i . e., l ∈ L o , thus x 1 = λ ol ( l 1 ) , x 2 = λ or ( l 2 ) , and x 3 = λ or ( l 3 ) . The decim al format o f l 1 , l 2 , and l 3 are 6 , 15 , and 7 , respectiv ely . In Fig. 4(d), it can be obs erved that among the shaded symbols that λ ol operates on, sy mbol S 16 is mapped by decimal label 6 . As a resul t , x 1 = S 16 . Considering the m apping function λ or indicated i n Fig. 4(c) we also have x 2 = λ or (15) = S 14 and x 3 = λ or (7) = S 12 . Consequently , l is mapped to x = ( S 16 , S 14 , S 12 ) . B. P erformance Comparis o n The proposed mappings are compared to the MD mapp ings obtained us i ng the state of the art methods in the literature such as the optimum m apping method in [11], the BSA, random mapping, and the MD m apping method in [12]. T o assess differe nt mappi ngs, we ﬁrst compare the value of Φ( µ, χ ) and ˆ Φ( µ, χ ) offere d b y the mappings. Then, we use the BER curve to compare the BICM-ID error performance in the low SNR region when using d iff erent m app i ngs. Finally , we use an analytical bound on the error -ﬂoor to ev aluate the system’ s error performance in the high SNR region. In our simulations, we consi der a rate- 1 / 2 con volutional code wi th the generator polynomial of (13 , 1 5 ) 8 . An interlea ver l eng t h of about 10000 bits is used. All BER curves are presented with sev en iterations and all gains reported in this secti o n are measured at a BER of 10 − 6 . Al s o, the error -ﬂoor b ounds have been plott ed using t he Gauss-Chebyshev method i n [22]. It is worth noting that achieving these error-ﬂoor boun ds can be challenging because they happen at a very small value of BER. 19 T ABLE III C O M PA R I S O N O F Φ( µ, χ ) A N D ˆ Φ( µ, χ ) F O R D I FF E R E N T M A P P I N G S . Mapping N = 2 N = 3 Φ( µ, χ ) ˆ Φ( µ, χ ) Φ( µ, χ ) ˆ Φ( µ, χ ) Optimum MD 8 -PSK [11] 0.3112 3. 3 529 0.2119 3.5454 Proposed MD 8 -PSK 0.3112 3. 3 529 0.2119 3.5454 BSA MD optimum 8 -QAM 0.4506 2. 7 838 0.3002 2.8515 Proposed MD optimum 8 -QAM 0.4552 2.9697 0.3041 3.1463 According to the modulation order , we discuss our resul ts for th ree classes of m odulations: low , medium, and high o rder modulati ons. In particular , we discuss 8 -ary m odulations (for the low order), 16 - and 32 -QAM (for the medium order), and M-QAM ( M = 64 , 128 and 256 ) for the higher order modulation s . E b /N 0 (in dB) 2 3 4 5 6 7 8 BER 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Optimum 4D 8-PSK BSA 4D optimum 8-QAM Proposed 4D optimum 8-QAM Optimum 6D 8-PSK BSA 6D optimum 8-QAM Proposed 6D optimum 8-QAM Fig. 5. BER performance of BICM-ID with 4 -D and 6 -D 8 -ary modulations over Rayleigh fading channels. 1) MD mapping of low or der modulati ons: T able III compares the values of the harmonic mean before and after feedback, i.e., Φ( µ, χ ) and ˆ Φ( µ, χ ) , for d i f ferent 2 N -D ( N = 2 , 3 ) mappings of 8 -ary modulations. As this table sho ws, for 8 -PSK, our proposed MD mapping provides t he same values as those of the optimu m MD mapping p roposed in [11]. This clearly shows our proposed algorithm’ s efﬁcienc y . W e have also used our algo rithm to obtain MD mapp i ngs of the optim um 8 - QAM cons t ellation introduced in [27]. As s hown in T able III, for the o p timum 8 -QAM, ou r resulting 20 E b /N 0 (in dB) 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 BER 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 Optimum 4D 8-PSK [10] BSA 4D optimum 8-QAM Proposed 4D optimum 8-QAM Optimum 6D 8-PSK [10] BSA 6D optimum 8-QAM Proposed 6D optimum 8-QAM Fig. 6. Error-ﬂoor bounds of BER f or B I C M-ID with 4 -D and 6 -D 8 -ary modulations over Rayleigh fading channels. mappings i mprove the values of Φ( µ, χ ) and ˆ Φ( µ, χ ) in comparison with t h e mappin g s obt ai n ed using the state of the art BSA. Therefore, it is expected that for the optim um 8 -QAM, the proposed mappings o utperform the BSA mappings in both the low and high SNR regions. This is con ﬁrmed by the plotted simulation resul ts for the BER and error-ﬂoor bounds of BICM-ID in Fig. 5 and Fig. 6, respectively . As sh own in Fig. 5 , for the 4 -D and 6 -D opt i mum 8 -QAM, ou r proposed mappings outperform their BSA counterparts by 1 and 0 . 85 dB, respectiv ely . The gain over the optim um 4 -D and 6 -D mapping of 8 -PSK is 0 . 7 and 0 . 65 , respectively . T h i s is because in addition to a large value of ˆ Φ( µ, χ ) , the p rop osed M D mappi ng of the optimum 8 -QAM si g niﬁcantly improves the value of Φ( µ, χ ) compared to that of the optimum M D 8 -PSK. Note that for the case of MD 8 -PSK, our resulting mapping is the same as the optimum mappi ng in [11]. Thus, o n l y the optim um mapping is used for com p arison in BER and error-ﬂoor plots. As shown in Fig. 6, the p rop osed 4 -D and 6 -D mapping s o f the opt i mum 8 -QAM outperform the correspondi n g BSA m app i ngs by 0 . 3 and 0 . 45 d B, respectively . Moreover in th i s ﬁgure, the optim um M D mappi ng of 8 -PSK offer s a lower error -ﬂoor due to the nature of the 8 -PSK constellati on, wh i ch can o ffer higher values o f ˆ Φ( µ, χ ) . 21 E b /N 0 (in dB) 0 2 4 6 8 10 12 BER 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 BSA 4D 16-QAM Random 4D 16-QAM 4D 16-QAM Proposed 4D 16-QAM BSA 4D 32-QAM Random 4D 32-QAM Proposed 4D 32-QAM Fig. 7. BER performance of BICM-ID with 4 -D 16 - and 32 -QAM ov er Rayleigh fading channels. 2) MD mappi ng of mediu m or der modulation s : T able IV li sts the values of Φ( µ, χ ) and ˆ Φ( µ, χ ) for different 4 -D mapp i ngs of 1 6 - and 32 -QAM. It ca n be seen from th i s table that, except for the value of Φ( µ, χ ) for the 4 -D 16 -QAM where the proposed mapping o ffers the same v alue as that of the mapp i ng i n [12], our propos ed mapp i ngs sign i ﬁcantly im prove th e values of Φ( µ, χ ) and ˆ Φ( µ, χ ) , comp ared to their well-known counterparts. Therefore, it is expected that the proposed mappings improve the error rate performance com p ared t o the p reviously known mappings in both the low and h igh SNR regions. This is conﬁrmed by the simulation results for the BER p erformance of BICM-ID shown in Fig. 7 and by the error-ﬂoor bounds plotted in Fig. 8. In the case of 4 -D 16 -QAM, as illustrated in Fig. 7, in th e lo w SNR region, the proposed mapping achieves a gain T ABLE IV C O M PA R I S O N O F Φ( µ, χ ) A N D ˆ Φ( µ, χ ) F O R D I FF E R E N T M A P P I N G S . 4 D Mapping Φ( µ, χ ) ˆ Φ( µ, χ ) 16 -QAM BSA mapping 0.2026 2.5814 Random mapping 0.2012 1.4350 Mapping in [12] 0.2 1 51 2.8491 Proposed mapping 0.2151 3.1622 32 -QAM BSA mapping 0.1027 2.8574 Random mapping 0.1008 1.2567 Proposed mapping 0.1117 3.1677 22 E b /N 0 (in dB) 6 7 8 9 10 11 12 BER 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 BSA 4D 16-QAM Random 4D 16-QAM 4D 16-QAM [11] Proposed 4D 16-QAM BSA 4D 32-QAM Random 4D 32-QAM Proposed 4D 32-QAM Fig. 8. Error-ﬂoor bounds of BER f or B I C M-ID with 4 -D 16 - and 32 -QAM ov er R ayleigh fading channels. of 1 and 1 . 55 dB over the BSA and random mappings, respecti vely . Although as sho wn in Fig. 7 the BER performance of the prop o sed 4 -D 16 -QAM is si milar t o that of the mappi ng in [12], our mapping out performs th e mapping in [12] in the high SNR region by about 0 . 5 dB, as shown in Fig. 8. This ﬁgure also shows that t h e proposed mappi ng improves the error -ﬂoor by 0 . 9 and 3 . 4 dB compared to the BSA and random mapping s , respectively . In th e case of 4 -D 32 -QAM, ou r proposed mapping outperforms the best pre viously known mappings , i.e., the BSA and random mappings , by 1 . 4 and 1 . 9 dB, respectiv ely , which is illust rated in Fig. 7. Fig. 8 shows that the corresponding gain in the h igh SNR re gion is 0 . 5 and 4 dB, respectively . 3) MD mapping of higher or der modula t ions: T able V reports the values of Φ( µ, χ ) and ˆ Φ( µ, χ ) for MD mappi ngs of differe nt high er o rder modul ati ons. It can be observed from this table that our proposed mappings improve these values com p ared to the pre v i ously kno wn m appings, e xcept for the case of the MD 64 -QAM in [12], where o ur mapp i ng offers a si m ilar value of Φ( µ, χ ) . Therefore, it is expected that the p rop o sed mappings offer improved error performance i n both the lo w and high SNR regions. This is conﬁrmed by the sim u lation results for the BER and by the analytical error bounds in Fig. 9-12. In the case of 4 -D 64 -QAM, our proposed mapping out performs the BSA and random mappings in the low SNR re gion by 2 . 9 and 3 . 1 dB, respecti vely , as shown in Fig. 9. 23 E b /N 0 (in dB) 4 5 6 7 8 9 10 11 12 13 14 BER 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 BSA 4D 64-QAM Random 4D 64-QAM 4D 64-QAM Proposed 4D 64-QAM Fig. 9. BER performance of BICM-ID with 4 -D 64 -QAM ov er Rayleigh fading channels. Although i n Fig. 9 our proposed m apping offers similar p erformance to that of the m apping in [12], the prop o sed mapp i ng outperforms the m apping in [12] by over 0 . 5 dB in the high SNR re gion, as shown in Fig. 1 0 . In this ﬁgure, t he achieved gain over th e BSA and rando m mapping s is 0 . 75 and 4 . 3 d B, respectively . As sh own i n Fig. 11, for 4 -D 128 -QAM in the low SNR region, the proposed mapping achiev es the gain of 3 . 1 and 3 dB over the BSA and random m appings, respectiv ely . The corresponding gain for the case of 4 -D 256 -QAM is 3 . 5 and 3 . 2 dB, respectively . F or the case of 4 -D 128 -QAM in the high SNR region, our propos ed mappi ng improves the error-ﬂoor by 2 and T ABLE V C O M PA R I S O N O F Φ( µ, χ ) A N D ˆ Φ( µ, χ ) F O R D I FF E R E N T M A P P I N G S . 4 D M apping Φ( µ, χ ) ˆ Φ( µ, χ ) 64 -QAM BSA mapping 0.0481 2.6899 Random mapping 0.0478 1.1688 Mapping in [12 ] 0 . 0 579 2.8166 Proposed mapping 0.0568 3.1683 128 -QAM BSA mapping 0.0245 1.8566 Random mapping 0.0245 1.1430 Proposed mapping 0.0294 3.2273 256 -QAM BSA mapping 0.0118 1.1282 Random mapping 0.0120 1.1034 Proposed mapping 0.0144 3.2389 24 E b /N 0 (in dB) 8 9 10 11 12 13 14 BER 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 BSA 4D 64-QAM Ranodm 4D 64-QAM 4D 64-QAM [11] Proposed 4D 64-QAM Fig. 10. Error-ﬂoor bounds of BE R for BICM-ID with 4 -D 64 -QAM over R ayleigh fading channels. 4 . 5 dB compared to the BSA and random mappings, respectively , as il lustrated in Fig. 12. Th e corresponding gain for the case of 4 -D 256 -QAM is 4 . 5 and 4 . 6 dB, respecti vely . E b /N 0 (in dB) 6 8 10 12 14 16 18 20 BER 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 BSA 4D 128-QAM Random 4D 128-QAM Proposed 4D 128-QAM BSA 4D 256-QAM Random 4D 256-QAM Proposed 4D 256-QAM Fig. 11. BER performance of BICM-ID wit h 4 -D 128 - and 256 -QAM over R ayleigh fading channels. It can be observed from Fig. 5-12 that the BSA results become less efﬁcient as the modulation 25 E b /N 0 (in dB) 14 15 16 17 18 19 20 BER 10 -15 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 BSA 4D 128-QAM Ranodm 4D 128-QAM Proposed 4D 128-QAM BSA 4D 256-QAM Ranodm 4D 256-QAM Proposed 4D 256-QAM Fig. 12. Error-ﬂoor bounds of BE R for BICM-ID with 4 -D 128 - and 256 -QAM ov er R ayleigh fading channels. order increases. As such, random mapping performs better than th e BSA mapping for 4 -D 256 -QAM, as shown in Fig. 1 1. Fig. 13 plots the BER curv es for diffe rent MD mappi ngs of 1024 -QAM. As it can be seen from this ﬁgure, the proposed m apping s i gniﬁcantly o u tperforms th e other counterparts. It is als o worth noting that i n this ﬁgure, the rando m mapping outperforms the BSA mapping. Thi s is because when looking for a mapping for 4 -D 1024 -QAM, the BSA i s unabl e to ﬁnish on e rou n d of the algorithm in a reasonable time frame. As a resul t , the obt ained BSA mapping is not very suitabl e. 26 E b /N 0 (in dB) 5 10 15 20 25 BER 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 BSA 4-D 1024-QAM Random 4-D 1024-QAM Proposed 4-D 1024-QAM Fig. 13. BER performance of BICM-ID wit h 4 -D 1024 -QAM ov er Rayleigh fading channels. C. Analysis of Con ver gence Behaviour The extrinsic inform ation transfer chart (EXIT chart) [17] is a commonly used metric to assess the con ver gence behavior of BICM-ID. In an EXIT chart, the area between the d ecoder curve and the demapp er curve i s called EXIT tunnel [16]. BICM-ID can achiev e a coding gain th rou gh the iterativ e decoding process on ly when it provides an open EXIT tunnel. Fig. 14 shows the EXIT charts for BICM-ID when usi ng diff erent 4 -D mappings of 1 2 8 -QAM in the Rayleigh fading channel. For bre vity in this s ecti o n, we in ve stigate the EXIT chart only for 4 -D 128 -QAM. The results are similar for other considered mo dulations. From Fig. 14, it can be seen th at BICM-ID wit h the proposed mapping exhibits an open Exit tunnel wh en E b N 0 = 6 dB. This im plies that when using the propo s ed mapping, the iterati ve decodin g process starts to improv e the performance of BICM-ID at E b N 0 = 6 . This is i n accordance with the BER curve of the proposed mapping in Fig. 11 , where the BER curve falls graduall y after E b N 0 = 6 . Fig. 14 also shows th at the op en EX IT tun n els for the BSA and random mappings appear at E b N 0 = 11 . As a result, when usi n g the BSA and random mappings , the system BER will not i mprove unless after E b N 0 = 11 , which is conﬁrmed by the correspondi ng BER curves in Fig. 11. Cons equently , t he BER performance wit h our proposed mapping improves th rou gh the 27 iterativ e decoding 5 dB earlier than those of the BSA and Random mappings. This results in an earlier turbo cliff for th e BER curve with our proposed mapping i n Fig. 11. I e Dec , I a Map 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 I a Dec , I e Map 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Decoder BSA mapping, E b /N 0 = 6 dB Random mapping, E b /N 0 = 6 dB Proposed mapping, E b /N 0 = 6 dB BSA mapping, E b /N 0 = 11 dB Random mapping, E b /N 0 = 11 dB Proposed mapping, E b /N 0 = 11 dB Fig. 14. EXIT chart for different 4 -D mappings of 128 -QAM in the R ayleigh fading channel. V I . C O N C L U S I O N A novel MD mapping method i s proposed to improve th e error performance of the BICM-ID system in both the low and high SNR regions ove r Rayleigh fading channels. The method uses four 2 -D mapp i ngs t o construct an M D mapping that improves the error performance in the l ow SNR region. Furthermore, cost functions are de veloped and optimized over the 2 -D mappings to achie ve an MD mapping th at i mproves the err or perf ormance in the high SNR re gion. Due to the l ower complexity of the 2 -D space, the o ptimization approach is very sim ple and results in excellent M D mappin gs for different modulati ons, includ i ng higher order modulations such as 2 m - QAM ( m = 4 , .., 10 ). Ext ens iv e n umerical results, including analytical and sim ulation results, show that the ob t ained mappings signiﬁcantly out p erform the pre viously known st at e of the art mappings in both th e lo w and hi gh SNR regions. 28 A P P E N D I X A P RO O F O F P RO P O S I T I O N 1 Let i = ( i 1 , i 2 , · · · , i D ) be th e set of bit posi t ions at which l 1 and l 2 are dif ferent. Let us deﬁne o j and z j as o j = D X k =1 I ( l i k j = 1) , (29) z j = D X k =1 I ( l i k j = 0) , where j = 1 , 2 and I ( θ ) is an indicator functi on th at takes value one if θ i s true, otherwise it equals zero. Clearly , z 1 = o 2 and z 2 = o 1 . Let ¯ o b e the numb er o f bit positions in wh i ch both l 1 and l 2 represent the bit value one. As w 1 = o 1 + ¯ o and w 2 = o 2 + ¯ o , we can write W = w 1 + w 2 = ( o 1 + ¯ o ) + ( o 2 + ¯ o ) , (30) = o 1 + o 2 + 2 ¯ o, ( o 2 = z 1 ) = o 1 + z 1 + 2 ¯ o, = D + 2 ¯ o. In (30), 2 ¯ o is an e ven number; as a result, if W ∈ E , D ∈ E , and i f W ∈ O , D ∈ O . A P P E N D I X B P RO O F O F P RO P O S I T I O N 2 Let l and ˜ l be two mN -bit labels . If l , ˜ l ∈ L e or l , ˜ l ∈ L o , the corresponding v alue for W in Proposition 1 is an eve n number . As a result, l and ˜ l have an ev en Hamming distance from each other . Similarly , two labels one from L e and the other from L o hav e an odd Hamming distance from each other . Since there is n o common 2 -D s ignal point between χ el and χ ol , there is no common 2 N -D signal po int b etween χ e and χ o . As a result , none of the 2 N -D signal points w i ll b e mapped simultaneous l y by a label from L e and a label from L o . Therefore, it is sufﬁcient to p rove that there 29 is a one-to-one correspondence between labels from L e and si gnal points from χ e and similarly between labels in L o and sign al point s in χ o . In what follo ws, we prov e this for th e e ven s ubsets, i.e., for labels in L e and signal poi n ts in χ e . Assume that l = ( l 1 , l 2 , · · · , l mN ) and ˜ l = ( ˜ l 1 , ˜ l 2 , · · · , ˜ l mN ) are two l abels in L e and are mapped to x = ( x 1 , · · · , x N ) a nd ˜ x = ( ˜ x 1 , · · · , ˜ x N ) , r espectiv ely , where both x and ˜ x ar e in χ e . L et us deﬁne l i and ˜ l i as th e i th m -tuple bits of l and ˜ l , respectiv ely . Then l = ( l 1 , l 2 , · · · , l N ) and ˜ l = ( ˜ l 1 , ˜ l 2 , · · · , ˜ l N ) . Based on the relation between l i and ˜ l i for d iff erent values of i , there are two possible cases as follows: Case 1: There exists a v alue of i ( i > 2 ) such t hat l i 6 = ˜ l i . Let j > 2 , then according to (6), the same one-to-one mappi ng funct i on, i .e., λ er ( . ) , is used t o map l j to x j and ˜ l j to ˜ x j . Therefore, because l j 6 = ˜ l j , we have x j 6 = ˜ x j ⇒ x 6 = ˜ x . (31) Case 2: l i = ˜ l i for all i > 2 . In th i s case, l i 6 = ˜ l i only when i = 1 , and as a result, d H ( l , ˜ l ) = d H ( l 1 , ˜ l 1 ) . Since l and ˜ l belong to L e , t hey hav e an e ven H am ming distance from each other . Consequently , the Hamm i ng distance between l 1 and ˜ l 1 is ev en as well. Howe ver , the two labels that are mapped to each symb ol in χ el hav e an odd Hamm ing dist ance from each other . Therefore, because λ el ( l 1 ) 6 = λ el ( ˜ l 1 ) , we ha ve x 1 6 = ˜ x 1 ⇒ x 6 = ˜ x . (32) From (31) and (32), it is concluded that in the prop o sed mapping function, diffe rent labels from L e are mapped to differe nt signal p oints in χ e . In a simil ar way , it can be proven that the different labels from L o are mapped t o the different si gnal po i nts i n χ o . As a resul t, the propo s ed MD mappi ng function is bi jectiv e. 30 A P P E N D I X C P RO O F O F P RO P O S I T I O N 3 Similar to propo sition 1, assum e that l = ( l 1 , l 2 , · · · , l N ) and l ′ = ( l ′ 1 , l ′ 2 , · · · , l ′ N ) are t wo labels in L e and are mapped to x = ( x 1 , · · · , x N ) and x ′ = ( x ′ 1 , · · · , x ′ N ) , respectively , where both x and x ′ are i n χ e , and l i and l ′ i are t he i th m -tuple bits of l and l ′ , respecti vely . Let us assume that d H ( l , l ′ ) ≥ m + 2 and j i s the number of values for i such that l i 6 = l ′ i . Note that d H ( l , l ′ ) > m , thus j > 1 . Based on the va lue of j , there are two poss ible cases as follows. Case 1 (when j = 2 ): Assum e that l i 6 = l ′ i for i = p, q , where q > p > 1 . W e have d H ( l , l ′ ) = d H ( l p , l ′ p ) + d H ( l q , l ′ q ) . (33) Let p = 1 . Because l q and l ′ q are m -tuple vectors, d H ( l q , l ′ q ) ≤ m . Moreover , d H ( l , l ′ ) ≥ m + 2 . Thus, using (33) w e ha ve d H ( l p , l ′ p ) ≥ 2 . Since p = 1 , t he mapping function λ el in (6) is u s ed to map l p and l ′ p . Note t hat in λ el , two m -tuple labels with Hamming d istance more than one bit cannot be mapped to the same 2 -D sym bol. As a result, we h a ve λ el ( l p ) 6 = λ el ( l ′ p ) , and therefore, x p 6 = x ′ p . Similarly , since q > 1 , the mapping function λ er in (6) is used to map l q and l ′ q . Moreo ver , λ er in (6) maps different m -tuple labels to different 2 -D symb o ls. T h erefore, as l q 6 = l ′ q , λ er ( l q ) 6 = λ er ( l ′ q ) , and as a result, x q 6 = x ′ q . Let p > 1 . In this case, λ er is used to map l p , l ′ p , l q , and l ′ q . As l p 6 = l ′ p and l q 6 = l ′ q , λ er ( l p ) 6 = λ er ( l ′ p ) and λ er ( l q ) 6 = λ er ( l ′ q ) . As a result, x p 6 = x ′ p and x q 6 = x ′ q . Therefore , for all v alues of p and q , x and x ′ are different in more than one symbol . Case 2 (when j > 3 ): There are at l east two values, p and q , such that p > q > 1 and l i 6 = l ′ i when i = p, q . In this case, λ er is u sed to map l p , l ′ p , l q , and l ′ q . As l p 6 = l ′ p and l q 6 = l ′ q , λ er ( l p ) 6 = λ er ( l ′ p ) and λ er ( l q ) 6 = λ er ( l ′ q ) . As a result, x i 6 = x ′ i when i = p, q . Th erefore, x and x ′ are di f ferent in more than one s y mbol. Consequently , in the proposed MD m apping function, when the Hammin g dis t ance between tw o labels i n L e is l ar ger than ( m + 1 ) bits, t he corresponding s y mbols-vectors in χ e are differ ent i n more th an one symbol, and therefore, cannot be t he nearest neighbours. The same characteristic can 31 T ABLE VI P RO P O S E D λ er , λ or , λ el , A N D λ ol F O R 32 - Q A M . λ er [24 17 13 8 25 29 1 9 21 28 0 5 16 20 4 12, 30 22 6 2 23 18 14 7 26 1 9 15 11 27 31 3 10] λ or [7 2 30 22 6 14 18 23 10 15 19 26 3 11 27 31, 13 9 25 24 8 1 29 21 4 0 28 20 5 12 17 16] λ el [(13 29) (8 24) (1 17) (9 25) (0 16) (5 21) (4 20) (12 28), (6 22) (2 18) (14 3 0) (7 23) (15 31) (11 27) (3 19 ) (10 26)] λ ol [(7 23) (2 18) (6 22) (14 30 ) (10 26) (15 31) (3 19) (11 27), (13 29), (9 25) (8 24) (1 17) (4 2 0) (0 16) (5 21) (12 28)] be proven for t he labels in L o and the correspond i ng symbol-vectors in χ o . A P P E N D I X D P RO O F O F P RO P O S I T I O N 4 Since y i > 0 for all i , then 1 P N i =1 y i 6 1 y j , j = 1 , 2 , · · · , N . (34) By taking the s ummation over all values of j from both si des of (34), the inequalit y in (12) can be written as 1 P N i =1 y i 6 1 N N X j =1 1 y j . (35) A P P E N D I X E P RO P O S E D M A P P I N G S 32 T ABLE VII P RO P O S E D λ er , λ or , λ el , A N D λ ol F O R 64 - Q A M . λ er [63 55 61 53 48 56 50 58 62 54 46 38 35 4 3 51 59, 60 52 36 37 32 33 49 57 47 39 44 45 40 41 34 42, 21 6 4 5 0 1 3 16 14 29 1 2 13 8 9 24 1 1 , 22 30 28 20 17 25 27 19 23 31 15 7 2 10 26 18] λ or [0 8 17 2 7 20 13 5 1 9 25 10 15 28 12 4, 16 24 27 26 31 30 29 21 3 11 19 18 23 22 14 6, 49 57 59 58 63 62 60 52 34 42 51 50 55 54 47 39, 41 33 56 48 53 61 36 44 40 32 43 35 38 46 37 45] λ el [(14 46) (6 38) (4 36) (5 37) ( 0 32) (1 33) (3 35) (11 43), (30 62) (21 53) (12 44) (13 45) (8 40) (9 41) (16 4 8) (27 59), (22 54) (29 61) (20 52) (7 39) (2 34) (17 4 9 ) (24 56) (19 51), (23 55) (31 63) (28 60) (15 47) (10 42) (25 57) (26 58) (18 50)] λ ol [(0 32) (8 40) (17 49) (2 34) (7 39) (20 52) (13 45) (5 37 ), (1 33) (16 48) (25 57) (10 42) (15 47) (28 60) (21 53) (4 36), (9 41) (24 56) (27 59) (26 58) (31 63) (22 54) (29 61) (12 44), (3 35) (11 43) (19 51) (18 50) (23 55) (30 62) (14 46) (6 38)] T ABLE VIII P RO P O S E D λ er , λ or , λ el , A N D λ ol F O R 128 - Q A M . λ er [100 103 37 86 105 122 32 115 116 11 8 36 70 104 107 106 113, 102 110 126 111 127 96 114 9 8 119 101 108 124 109 82 66 99, 117 94 78 125 64 67 112 97 87 76 79 95 80 72 83 74, 68 71 92 77 88 65 75 90 61 84 69 93 73 8 9 91 120, 60 31 13 85 24 8 81 121 14 15 12 29 25 11 16 3, 30 5 21 28 9 2 7 17 19 52 23 7 20 57 56 26 1, 39 4 22 59 43 10 49 51 38 54 6 41 58 40 42 3 3 , 53 46 47 63 18 2 50 48 62 55 44 45 0 123 34 35] λ or [17 24 121 0 44 61 85 15 16 8 81 3 60 31 13 14, 25 9 27 56 20 21 28 29 11 26 40 59 57 4 5 12, 1 1 0 49 4 3 41 22 7 30 19 48 58 42 6 5 2 53 23, 18 35 51 33 54 39 55 47 123 2 50 34 38 46 62 37, 107 122 32 98 102 118 36 63 104 106 115 114 126 119 70 86, 113 97 99 96 110 1 00 103 116 83 112 8 2 66 111 101 7 8 117, 72 67 64 109 127 108 94 76 65 88 80 125 124 95 79 77, 74 89 90 120 68 71 69 87 75 73 91 105 45 84 93 92] λ el [(37 101) (32 96) (36 100) (61 125) (60 124) (31 95) (13 77) (24 88), (8 72) (14 78) (15 79) (12 76) (29 93) (25 89) (11 75) (16 80), (3 67) (30 94) (5 69) (21 85) (28 92) (9 73) (27 91) (17 8 1), (19 83) (52 116) (23 8 7 ) (7 71) (20 84) (57 121) (56 120) (26 90), (1 65) (39 103) (4 68) (22 86) (59 123) (43 107) (10 74) (49 113), (51 115) (38 102) (54 118) (6 70) (41 105) (58 122) (40 104) (42 106), (33 97) (53 117) (46 110) (47 111) (63 127) (18 82) (2 66) (50 114), (48 112) (62 126) (55 119) (44 108) (45 109) (0 64) (34 98 (35 9 9)] λ ol [(17 8 1 ) (24 88) (0 64) (45 109) (60 124) (15 79) (16 80) (8 72), (3 67) (63 127) (31 95) (13 77) (14 78) (25 89) (11 75) (9 73), (27 91) (57 121) (21 8 5 ) (28 92) (29 93) (56 120) (10 74) (40 104), (59 123) (20 84) (4 68) (7 71) (12 76) (26 90) (1 65) (49 113), (43 107) (41 105) (22 86) (23 87) (5 69) (19 83) (48 112) (51 115), (58 122) (6 70) (39 103) (52 116) (30 94) (18 82) (50 114) (35 99), (42 106) (54 118) (55 119) (62 126) (53 117) (2 66) (34 98) (33 97), (38 102) (46 110) (37 101) (47 111) (32 96) (36 100) (44 108) (61 125)] 33 T ABLE IX P RO P O S E D λ er , λ or , λ el , A N D λ ol F O R 256 - Q A M . λ er [206 205 222 237 236 220 252 211 227 195 194 135 215 204 231 199, 207 254 2 21 141 253 200 1 4 0 243 167 226 15 1 210 247 83 230 198, 238 142 2 01 88 217 157 216 1 56 13 1 20 9 242 166 19 3 134 213 197, 239 255 89 158 17 3 232 172 18 8 163 130 183 225 208 150 246 214, 223 218 1 74 233 249 189 2 4 8 136 162 147 24 1 129 165 81 245 229, 202 219 1 59 191 190 137 1 5 3 152 179 145 14 6 182 224 149 192 133, 203 234 2 50 138 154 169 1 8 5 168 177 161 24 0 144 181 80 244 212, 143 175 2 51 170 155 187 1 8 6 184 178 176 16 0 128 164 148 132 228, 91 235 139 60 171 25 2 4 56 49 17 51 180 35 3 115 99, 92 124 44 28 120 40 57 59 48 16 50 34 19 2 39 67, 12 72 29 61 121 8 41 58 32 33 113 18 55 114 98 7, 108 125 73 104 62 9 27 26 52 112 1 54 9 7 38 23 66, 93 109 13 45 105 63 10 42 43 0 53 96 22 65 82 103, 77 94 90 30 46 122 123 11 36 20 37 21 5 6 119 196, 76 95 126 14 127 31 106 47 116 4 117 64 118 102 85 71, 78 79 110 111 74 75 15 1 07 84 101 100 68 69 86 70 87] λ or [49 48 16 32 11 2 0 52 36 43 42 27 26 58 59 57 56, 17 33 1 53 21 117 20 116 47 11 123 10 63 41 40 24, 50 113 54 96 3 7 64 84 4 107 106 46 122 105 62 9 25, 51 18 97 65 22 5 101 100 15 1 2 7 31 90 3 0 104 121 8 , 34 55 114 38 6 118 69 68 75 74 14 13 73 45 61 120, 19 2 23 82 119 102 86 111 110 95 126 109 125 72 29 60, 35 3 98 7 66 103 85 70 79 77 94 108 12 124 44 28, 180 115 39 99 67 196 71 87 78 76 93 92 91 235 139 171, 80 148 244 22 8 212 229 197 19 8 206 207 239 203 143 234 175 170, 164 149 1 32 133 246 214 2 3 1 199 205 238 22 3 142 202 219 251 138, 128 224 1 92 245 213 230 2 1 5 204 222 237 22 1 255 218 159 250 155, 181 81 150 193 134 210 247 135 220 236 254 141 174 89 191 154, 240 165 2 25 208 151 83 194 195 211 200 253 201 158 233 249 187, 160 144 1 29 183 166 209 2 2 6 227 252 140 21 7 173 88 190 137 169, 176 161 1 82 241 130 242 1 6 7 131 243 216 15 7 232 189 153 185 186, 178 177 1 79 146 145 162 1 4 7 163 156 188 17 2 248 136 152 168 184] λ el [(83 211) (88 2 1 6) (89 217) (81 209) (80 208) (91 219) (60 188) (25 15 3), (24 152) (56 184) (49 1 77) (17 145) (51 179) (35 163) (3 131) (115 243), (99 227) (92 220) (12 4 252) (44 172) (28 156) (120 248) (40 168) (57 185) , (59 187) (48 176) (16 1 44) (50 178) (34 162) (19 147) (2 130) (39 167), (67 195) (12 140) (72 2 00) (29 157) (61 189) (121 249) (8 13 6 ) (41 169) , (58 186) (32 160) (33 1 61) (113 2 41) (18 146) (55 183) (114 24 2) (98 226), (7 135) (108 236) (125 2 5 3) (73 201) (104 23 2 ) (62 190) (9 1 3 7) (27 155), (26 154) (52 180) (11 2 240) (1 129) (54 182) (97 225) (38 166) (23 151), (66 194) (93 221) (10 9 237) (13 141) (45 173) (105 233) (63 191) (10 138), (42 170) (43 171) (0 128) (53 181) (96 224) (22 150) (65 193) (82 210), (103 231) (77 205) (94 222) (90 218) (30 158) (46 174) (122 250) (123 251), (11 139) (36 164) (20 1 48) (37 165) (21 149) (5 133) (6 1 34) (119 247), (76 204) (95 223) (12 6 254) (14 142) (127 255) (31 159) (106 23 4 ) (47 175), (116 244) (4 132) (117 245) (64 192) (118 246 ) (102 230 ) (85 213) (71 199), (78 206) (79 207) (11 0 238) (111 239) (74 202) (75 203) (15 143) (107 235), (84 212) (101 229) (100 228) (68 196) (69 197) (86 214) (70 198) (87 215)] λ ol [(49 177) (48 1 7 6) (32 160) (112 24 0 ) (0 128) (53 1 8 1) (43 171) (52 180), (11 139) (10 138) (42 1 70) (26 154) (27 155) (58 186) (59 187) (56 1 84), (17 145) (33 161) (1 129) (54 182) (96 224) (21 149) (20 148) (36 164), (116 244) (122 250) (123 2 5 1) (63 191) (9 137) (41 169) (57 185) (16 144), (113 241) (97 225) (37 165) (117 245) (84 212) (4 132) (47 175) (31 15 9 ), (46 174) (30 158) (10 5 233) (62 190) (8 136) (24 152) (50 178) (18 146), (22 150) (5 133) (64 192) (101 229) (107 235) (106 2 3 4) (75 203) (127 255), (13 141) (45 173) (10 4 232) (121 249) (25 153) (51 179) (55 183) (38 166), (65 193) (6 134) (11 8 246) (68 196) (10 0 228) (15 143) (74 202) (14 142), (90 218) (73 201) (61 1 89) (40 168) (34 162) (2 130) (114 242 ) (23 151), (119 247) (102 230) (86 214) (69 197) (111 239) (110 238) (126 254) (109 237), (125 253) (72 200) (29 157) (120 248) (19 147) (115 243) (98 226) (82 210), (66 194) (103 231) (85 213) (70 198) (79 207) (95 223) (77 205) (94 222), (12 140) (44 172) (28 1 56) (35 163) (3 131) (39 167) (99 227) (7 135), (83 211) (71 199) (87 2 15) (78 206) (76 204) (93 221) (108 236) (92 220), (91 219) (124 252) (60 188) (80 208) (81 209) (67 195) (89 217) (88 216)] 34 T ABLE X P RO P O S E D λ er A N D λ or F O R 512 - Q A M . λ er [397 49 3 479 462 477 456 335 269 162 165 455 502 468 453 439 503, 411 334 509 494 461 472 488 285 374 342 372 402 4 8 5 501 434 385, 172 440 408 510 495 504 491 473 262 451 498 466 4 5 2 406 386 326, 475 174 409 511 492 507 271 366 164 481 449 486 4 7 1 390 340 260, 463 415 399 446 398 447 410 430 420 422 464 438 3 8 8 405 389 484, 414 412 396 474 458 442 428 394 423 421 448 384 4 8 7 436 437 404, 413 445 459 431 270 395 429 427 416 418 480 432 4 0 1 400 496 407, 444 506 443 393 489 286 424 426 417 483 261 358 4 3 3 403 465 391, 392 350 284 457 380 382 300 425 419 263 327 309 2 7 7 499 482 497, 490 505 268 173 318 301 282 264 259 257 256 258 3 0 8 356 435 387, 287 348 302 170 319 365 280 266 295 359 294 167 2 7 9 467 166 276, 364 317 316 281 299 330 314 298 275 288 304 273 1 6 0 292 310 324, 441 332 267 346 315 313 312 296 291 289 320 305 3 0 6 272 293 325, 303 383 265 362 378 10 361 297 355 290 352 3 7 0 336 353 311 274, 283 171 328 360 169 329 381 376 371 307 354 321 3 163 369 368 , 367 379 347 8 331 344 345 377 35 339 323 375 337 0 32 2 373, 168 363 58 11 42 333 40 41 19 39 32 33 8 17 48 161 2, 74 26 27 45 43 24 56 57 50 34 33 38 96 16 49 1, 126 44 123 59 122 9 25 120 11 5 99 113 10 3 97 102 53 240, 62 90 46 106 107 104 105 121 114 51 67 65 112 64 71 227, 201 91 47 124 75 72 125 73 83 98 18 8 1 36 80 52 7, 12 28 109 29 61 60 127 8 9 82 66 1 19 37 23 20 211 21, 219 251 31 92 249 110 111 88 117 118 55 100 243 6 241 209, 203 153 185 94 76 108 77 93 101 87 69 54 70 68 179 147, 136 253 235 349 217 13 248 95 85 116 22 225 193 226 4 129, 222 351 15 184 78 232 2 0 0 79 86 242 84 146 195 178 215 181, 191 220 236 255 152 239 252 216 194 210 245 244 2 4 6 198 148 130, 143 159 223 204 206 238 254 207 213 197 229 212 1 9 6 214 149 133, 234 137 218 139 508 157 237 14 177 208 183 1 2 8 470 199 230 5, 187 205 189 478 476 154 186 30 231 341 192 1 5 1 150 180 228 145, 155 156 140 142 141 202 250 63 224 176 144 4 5 4 500 134 131 247, 221 188 190 158 460 138 175 233 357 278 450 135 4 6 9 182 132 343] λ or [48 37 96 33 306 64 5 308 18 6 60 90 441 41 123 45 233, 225 112 113 339 161 129 224 240 250 26 10 42 40 107 6 1 127, 179 103 39 32 17 241 341 69 126 74 58 347 169 59 11 235, 343 16 0 1 65 81 80 21 6 3 124 122 43 345 171 10 6 44, 35 49 51 99 98 34 114 50 89 25 73 9 105 120 121 57, 163 3 115 53 83 66 2 18 72 185 88 249 91 75 104 56, 97 67 117 119 55 116 82 54 93 13 77 232 111 8 125 109, 19 227 1 3 1 102 2 26 38 118 52 15 3 248 95 29 137 2 4 108 27, 101 100 36 195 178 87 2 4 2 6 22 217 79 201 184 76 47 349, 243 68 85 23 84 70 194 86 216 152 2 0 0 136 15 21 9 187 251, 7 211 245 4 247 20 130 146 221 205 155 253 189 92 139 110, 193 147 229 183 230 215 210 246 157 223 236 239 2 5 5 237 12 203, 71 231 1 8 1 199 2 44 198 150 182 2 04 207 252 141 78 28 31 351, 176 144 135 197 213 151 134 214 159 220 238 222 2 5 4 94 202 234, 145 128 192 228 149 196 212 148 206 143 140 191 1 8 8 218 14 46, 310 209 342 208 133 166 132 180 156 158 142 174 1 5 4 30 138 62, 374 450 278 438 471 406 164 470 478 479 476 477 1 9 0 172 397 472, 262 402 466 502 390 454 468 404 462 415 412 460 4 1 3 461 408 456, 434 386 326 340 407 500 469 452 414 350 463 399 3 9 6 492 175 392, 279 439 276 422 486 455 405 464 398 510 494 511 5 0 8 495 411 509, 358 167 503 436 391 453 388 484 446 430 474 444 4 2 8 445 348 332, 387 260 418 501 437 400 448 389 286 410 447 284 4 9 3 287 475 443, 165 467 324 485 487 423 384 420 394 458 431 334 3 9 5 268 424 303, 419 356 435 403 277 496 421 401 270 382 490 429 4 5 9 366 507 267, 357 481 272 327 449 432 465 318 426 442 506 380 1 7 0 491 383 425, 293 295 160 336 263 385 480 416 316 302 319 300 4 2 7 301 296 365, 353 288 256 359 325 497 433 261 282 266 346 317 3 3 0 315 312 297, 305 289 352 321 257 273 320 417 298 314 378 362 2 9 9 376 379 377, 451 292 373 371 354 309 375 372 457 173 367 281 2 6 5 344 440 271, 499 368 307 369 322 258 311 482 285 488 393 328 3 2 9 331 280 364, 483 304 291 323 290 338 274 294 335 504 168 505 3 6 3 360 264 489, 275 259 337 355 370 177 162 498 409 473 269 333 3 6 1 313 381 283] 35 T ABLE XI P RO P O S E D λ el A N D λ ol F O R 512 - Q A M . λ el [(172 4 28) (168 42 4) (162 418) (165 421) (175 431) (173 429) (164 420) (170 42 6 ), (160 416) (167 423) (161 417) (42 298) (43 299) (169 425) (41 297) (57 313), (35 291) (33 28 9 ) (32 288) (1 257) (64 3 20) (177 43 3) (10 266) (58 314), (171 427) (122 378) (59 315) (123 379) (40 296) (121 377) (51 307) (49 305), (163 419) (96 352) (65 321) (17 273) (24 1 49 7 ) (80 336) (90 346) (74 330), (106 362) (45 301) (107 363) (56 31 2) (105 361) (120 376) (34 29 0 99 355), (97 353) (113 369) (37 2 9 3) (39 295) (103 359) (69 32 5) (124 380) (235 491), (61 317) (11 26 7 ) (109 365) (75 331) (104 360) (9 265) (50 306) (98 354), (115 371) (19 275) (48 304) (112 368) (81 337) (7 263) (63 319) (44 300), (127 383) (233 489) (125 381) (91 347) (73 329) (25 281) (89 345) (114 370), (3 259) (67 323) (225 481) (0 256) (16 272) (100 356) (139 395) (110 366), (47 303) (27 28 3 ) (108 364) (8 264) (88 344 ) (72 3 28) (66 322) (83 339), (53 309) (227 483) (101 357) (243 499 ) (179 435) (71 327) (12 268) (92 348), (251 507) (24 280) (111 367) (249 505) (185 44 1 ) (2 25 8 ) (117 373) (116 372), (119 375) (195 451) (131 387) (68 324) (211 467) (189 445) (219 475) (76 33 2), (187 443) (137 393) (232 488) (77 333) (93 349) (18 274) (82 338) (55 3 1 1), (226 482) (102 358) (85 341) (4 260) (229 485 ) (255 511) (253 50 9 ) (15 271), (201 457) (29 285) (95 351) (13 269) (54 310) (242 498) (118 374) (38 2 9 4), (87 343) (23 27 9 ) (36 292) (247 503) (230 486) (252 508 ) (239 49 5 ) (155 411 ), (136 392) (184 440) (79 335) (248 504) (153 409) (52 3 0 8) (6 262) (17 8 434), (70 326) (84 34 0 ) (20 276) (244 500) (198 454) (238 494 ) (207 46 3 ) (141 397 ), (205 461) (152 408) (200 456) (216 472) (21 7 4 7 3) (86 342) (22 278) (194 45 0 ), (130 386) (210 466) (215 471) (150 406) (21 2 4 6 8) (156 412 ) (159 415) (206 462), (220 476) (236 492) (204 460) (223 479) (22 1 4 7 7) (157 413 ) (146 402) (246 502), (182 438) (134 390) (214 470) (148 404) (18 0 4 3 6) (203 459 ) (94 350) (154 410), (142 398) (174 430) (14 270) (46 302) (234 490) (21 277) (176 432 ) (144 40 0 ), (208 464) (133 389) (149 405) (147 403) (19 3 4 4 9) (31 287) (78 334) (188 44 4 ), (158 414) (218 474) (138 394) (250 506) (12 6 3 8 2) (129 385 ) (128 384) (228 484), (166 422) (197 453) (231 487) (28 284) (254 510) (222 478) (140 396) (190 446), (202 458) (186 442) (26 282) (5 261) (240 496 ) (209 465) (135 39 1 ) (196 452 ), (213 469) (199 455) (245 501) (237 493) (19 1 4 4 7) (143 399 ) (30 286) (62 318), (60 316) (224 480) (145 401) (192 448 ) (132 388) (151 407) (183 439) (181 437)] λ ol [(225 4 81) (19 275) (163 419) (33 2 89) (64 320) (177 433) (60 316) (58 314), (42 298) (57 31 3 ) (40 296) (56 312) (125 381) (179 435) (48 304) (113 369), (49 305) (161 417) (17 2 7 3) (241 497 ) (224 480) (126 382) (10 266) (90 346), (43 299) (41 29 7 ) (123 379) (45 301) (103 359) (112 368 ) (37 293) (97 353), (1 257) (65 321) (80 336) (129 385) (234 490) (26 282) (74 330) (171 427), (169 425) (107 363) (11 267) (127 383) (69 325) (0 256) (39 295) (32 288), (96 352) (81 33 7 ) (63 319) (124 380) (106 362) (122 378 ) (59 315) (61 317), (235 491) (44 300) (35 291) (34 290) (51 307) (98 354) (11 4 3 7 0) (66 322), (89 345) (50 30 6 ) (185 441) (72 328) (25 281) (73 329) (9 265 ) (120 37 6 ), (121 377) (99 355) (3 259) (227 483 ) (53 309) (83 339) (55 311) (82 338), (18 274) (2 258) (249 505) (88 344) (111 367) (8 264) (91 347) (104 36 0 ), (105 361) (115 371) (67 323) (117 373) (116 372) (38 2 9 4) (242 498) (118 374), (13 269) (93 34 9 ) (77 333) (232 488) (137 393) (24 2 8 0) (27 283) (109 365), (75 331) (101 357) (119 375) (226 482 ) (102 358) (178 434) (52 308) (153 409), (54 310) (248 504) (29 2 8 5) (95 351) (184 440) (108 364) (47 303) (233 489), (243 499) (131 387) (195 451) (23 279) (70 326) (87 343) (6 262) (194 45 0), (217 473) (79 335) (201 457) (15 27 1) (76 332) (187 443) (16 272) (36 292), (85 341) (211 467) (84 3 4 0) (130 386 ) (146 402) (86 342) (22 278) (200 456), (216 472) (136 392) (219 475) (12 268) (92 348) (251 507) (10 0 35 6) (71 327), (68 324) (4 260) (247 503) (20 276) (210 466) (157 413) (221 477) (152 408), (155 411) (253 509) (189 445) (237 493) (13 9 3 9 5) (110 366 ) (7 263) (193 449), (229 485) (245 501) (244 500) (215 471) (13 4 3 9 0) (246 502 ) (205 461) (236 492), (141 397) (239 495) (255 511) (28 284) (31 287) (203 459) (5 261) (21 277), (181 437) (199 455) (230 486) (198 454) (15 0 4 0 6) (182 438 ) (223 479) (220 476), (207 463) (252 508) (254 510) (78 334) (94 350) (250 506) (24 0 49 6) (147 40 3), (231 487) (197 453) (183 439) (151 407) (21 2 4 6 8) (214 470 ) (204 460) (206 462), (238 494) (191 447) (188 444) (218 474) (14 270) (46 302) (17 6 43 2) (128 38 4), (208 464) (149 405) (213 469) (196 452) (13 2 3 8 8) (148 404 ) (159 415) (156 412), (222 478) (142 398) (190 446) (154 410) (20 2 4 5 8) (186 442 ) (209 465) (144 400), (228 484) (135 391) (133 389) (166 422) (16 4 4 2 0) (180 436 ) (143 399) (140 396), (158 414) (174 430) (30 286) (138 394) (62 318) (145 401) (192 448) (167 423), (172 428) (175 431) (165 421) (173 429) (17 0 4 2 6) (160 416 ) (162 418) (168 424)] 36 T ABLE XII P RO P O S E D λ er F O R 1024 - Q A M . λ er [674 195 226 193 197 229 67 225 129 16 3 161 165 97 69 203 65, 182 54 2 5 2 222 220 180 118 176 86 151 1 4 8 150 247 246 214 212, 194 199 227 231 131 133 200 99 23 4 202 37 101 1 3 3 235 201, 158 156 254 22 52 470 23 20 1 1 6 183 144 146 215 242 244 213, 192 228 224 162 66 64 3 5 35 45 1 483 139 485 481 449 2 33, 406 190 404 407 502 223 48 50 468 112 178 87 240 245 210 208, 196 230 130 167 205 237 450 487 453 389 385 77 72 171 137 169, 438 188 94 436 400 159 191 500 1 6 255 114 179 181 84 243 694, 198 128 98 68 232 39 448 141 13 8 419 355 421 74 417 75 7 3 , 126 30 1 2 4 432 434 402 92 184 496 18 248 119 8 2 147 149 662, 134 135 103 96 239 0 136 173 104 109 13 106 323 353 107 1 0 5, 62 278 2 8 342 374 127 95 186 471 498 503 55 221 80 145 209, 132 160 71 2 455 482 170 423 359 322 4 5 10 11 325 357 321, 60 310 30 4 476 31 368 370 439 403 435 152 464 25 3 19 115 211, 166 164 7 32 480 387 168 324 320 8 456 257 461 49 1 9 41, 412 510 306 308 274 276 372 343 336 340 189 157 51 250 177 241, 70 100 4 34 452 418 327 356 495 490 42 261 4 93 293 43 457, 446 508 478 272 279 307 63 56 375 371 120 154 466 218 216 83, 102 204 207 484 386 76 354 2 6 0 40 291 488 458 289 459 393 489 , 414 444 447 408 479 511 275 472 58 338 122 405 499 469 21 1 17, 238 36 1 4 3 175 384 79 111 47 259 295 397 42 9 427 363 361 425, 284 286 415 440 442 410 311 504 506 24 125 401 155 467 53 85, 6 486 45 4 420 12 15 352 292 258 392 426 394 395 301 331 329, 318 380 314 348 445 413 509 305 61 26 433 437 187 501 49 113, 236 140 391 416 78 326 263 256 4 63 431 424 365 269 328 265 297, 316 280 312 376 443 411 309 477 474 273 29 93 90 88 251 17, 38 206 38 8 108 358 294 460 290 288 399 26 8 367 271 36 0 299 267, 287 350 383 315 282 378 441 475 277 507 27 339 123 497 219 8 1, 673 142 172 422 14 44 494 492 428 270 36 4 332 303 264 266 330, 382 319 317 283 313 379 346 409 59 373 341 369 185 153 249 217, 581 174 390 110 46 262 396 398 4 30 462 302 300 333 362 298 296, 351 285 381 281 347 344 377 473 505 25 337 121 91 46 5 702 566, 609 745 961 681 585 617 833 553 521 969 335 873 366 777 334 809, 829 831 349 828 830 345 798 924 57 572 574 542 89 63 8 918 534, 577 713 993 649 837 869 865 555 805 1001 939 843 937 875 813 811, 825 827 792 826 796 892 862 956 958 926 822 790 636 700 670 764, 677 747 997 619 587 835 523 971 1003 905 845 841 781 815 779 810, 793 797 824 799 794 894 952 927 816 818 1022 540 944 950 668 766, 579 513 545 683 933 525 522 773 1005 941 907 877 842 776 778 808, 893 891 795 895 888 860 959 954 1020 820 990 543 606 916 1014 732, 741 715 651 586 929 621 557 968 973 769 801 840 879 783 812 780, 953 859 856 863 957 890 920 784 788 786 791 854 912 948 564 734, 611 613 549 584 897 589 970 554 807 1000 909 906 872 874 876 814, 921 889 858 955 923 922 823 991 575 988 882 886 919 671 560 688, 641 547 963 995 901 871 867 520 1007 803 943 938 904 936 847 782, 861 102 1 821 989 925 984 1016 819 787 880 639 914 946 735 982 630, 705 714 965 650 931 618 834 1002 552 771 975 844 768 911 940 878, 857 987 789 817 573 986 1018 570 56 8 884 604 1008 703 980 532 656, 737 578 517 999 685 616 836 832 866 559 772 770 775 1004 908 846, 910 101 7 1019 7 8 5 536 10 2 3 887 852 855 915 607 983 1012 562 535 660, 675 746 515 962 749 653 935 899 839 868 804 972 802 800 974 942, 985 569 539 538 541 883 850 848 947 951 696 1010 767 624 628 663, 709 643 712 717 744 682 960 996 623 591 864 527 838 1006 556 774, 885 537 571 851 945 637 634 632 669 664 698 528 567 631 598 690, 739 707 645 551 514 994 648 680 687 898 588 870 524 806 526 558, 849 853 881 605 913 600 917 976 701 733 765 530 596 599 658 692, 679 704 610 576 615 751 967 512 930 964 655 896 903 590 620 633, 601 603 635 602 949 667 666 978 760 1015 531 762 626 752 727 695, 706 642 740 580 583 546 612 516 992 719 966 932 928 900 902 622, 977 697 665 1009 699 979 981 728 730 563 592 659 720 722 754 759, 738 736 743 676 708 608 644 519 544 548 716 998 934 652 654 684, 761 729 1013 731 529 1011 565 53 3 627 594 657 661 691 724 756 726 , 678 672 742 710 711 646 640 647 582 614 550 518 718 748 686 750, 593 625 763 561 597 629 595 721 689 753 723 725 755 757 693 758] 37 T ABLE XIII P RO P O S E D λ or F O R 1024 - Q A M . λ or [829 825 827 797 793 893 891 889 856 859 921 861 9 87 857 1017 985, 910 846 942 908 940 878 782 876 814 783 812 778 808 810 811 809, 831 792 824 795 953 957 955 858 1021 821 789 1019 571 569 537 849, 774 556 1006 974 802 911 1004 844 84 7 936 874 780 776 815 779 334, 349 826 799 895 863 923 817 573 989 541 785 539 885 881 853 633, 558 526 806 524 972 800 775 772 768 904 938 872 879 781 875 813, 828 796 794 890 925 1016 984 986 538 536 945 851 605 635 603 697, 601 590 620 870 838 527 864 804 770 975 771 943 840 842 843 777, 345 888 860 922 823 1023 570 887 883 637 913 600 602 1009 665 902, 622 900 928 903 896 588 623 866 868 559 554 803 906 941 939 366, 830 894 920 991 819 1018 848 850 632 917 949 667 699 731 1013 977, 654 684 934 932 655 898 591 899 839 832 552 807 1000 909 907 937, 892 959 954 575 787 568 855 634 947 976 666 979 1011 561 729 761, 686 652 966 719 964 687 930 680 935 836 1002 1007 970 801 845 877, 798 784 788 880 884 882 852 701 669 978 1015 728 981 565 529 593, 718 998 548 544 992 996 648 960 653 931 616 834 520 968 769 841, 927 1 020 791 543 639 607 915 951 664 765 73 3 730 533 595 763 625, 748 518 716 516 751 967 512 682 685 901 618 871 867 557 1005 873, 952 820 786 604 914 696 1 008 1010 980 760 531 762 563 627 629 597, 750 519 612 615 546 994 514 962 749 650 897 525 522 773 973 905, 862 818 988 886 912 703 698 528 530 567 596 659 592 594 657 721, 550 614 582 608 583 551 744 717 999 965 584 589 621 835 971 805, 816 1 022 990 854 946 671 983 1012 7 67 631 626 599 661 720 753 689, 644 647 640 676 580 610 712 515 517 549 995 586 929 865 523 1003, 924 822 540 636 919 948 735 560 532 624 752 658 691 722 725 723, 646 710 711 708 645 576 746 714 547 513 963 651 933 837 555 1001, 956 572 542 944 916 564 982 732 562 628 598 663 727 724 755 693, 742 743 740 642 707 578 705 611 613 997 545 683 587 619 521 969, 958 926 638 606 950 668 1 014 764 734 535 656 660 754 756 759 757, 736 672 679 704 709 643 641 579 747 715 993 961 649 585 869 553, 57 790 89 700 918 702 670 566 766 534 630 690 69 2 695 758 726, 678 738 706 739 675 737 677 673 581 609 713 745 681 617 833 398, 25 574 121 91 465 249 219 217 81 113 688 241 662 694 208 213, 194 674 196 230 198 166 741 6 38 577 142 390 172 110 262 430, 505 337 369 185 153 497 49 25 1 17 8 5 117 83 209 210 242 212, 195 192 199 228 134 132 70 10 2 236 238 140 174 388 14 46 396 , 473 341 93 123 90 467 501 53 218 115 177 145 211 245 247 214, 227 226 231 224 128 135 164 7 204 4 86 206 454 422 108 4 4 4 28, 373 27 339 437 187 88 46 9 2 1 19 216 80 149 243 240 215 244, 193 197 131 162 130 98 160 4 2 0 7 36 143 391 420 78 294 492, 59 507 433 371 401 155 499 51 250 82 181 147 146 144 148 246, 225 229 133 64 167 68 71 100 32 452 175 384 416 358 494 462, 409 277 29 125 122 405 466 253 248 221 179 84 183 178 176 151, 129 67 99 163 66 103 96 239 34 484 386 418 12 32 6 460 288, 475 305 273 26 120 15 4 157 464 55 119 114 8 7 1 16 118 8 6 1 50, 182 161 165 200 205 232 2 455 0 480 76 111 15 292 290 270, 377 477 474 24 338 43 5 189 498 503 16 255 468 112 220 22 180, 65 97 234 202 5 237 39 448 482 168 79 354 352 263 463 302, 346 309 61 506 375 340 186 152 471 18 50 23 20 5 2 22 2 54 , 69 1 101 37 35 3 48 7 136 170 423 356 260 495 256 399 335, 379 441 509 504 5 8 336 439 403 184 496 500 223 48 156 254 252, 235 203 33 483 453 450 138 141 387 327 320 40 259 258 431 364, 313 378 411 472 307 56 372 127 92 402 191 159 470 404 158 406, 201 449 485 451 385 389 173 355 104 324 47 291 295 392 332 300, 347 344 443 511 311 343 276 95 37 0 434 400 407 436 502 188 190, 233 481 139 171 421 109 419 106 359 322 490 488 426 367 268 303, 281 283 445 413 442 410 275 63 27 4 368 31 28 94 4 3 2 438 73 , 169 137 75 72 417 77 13 10 8 456 4 2 3 97 424 365 271 333, 381 315 282 376 415 479 440 408 279 476 374 342 310 124 30 126, 105 107 323 353 7 4 45 2 57 261 461 289 429 394 269 360 264 362, 285 317 383 312 314 348 380 447 272 508 308 304 306 60 278 62, 321 357 9 3 2 5 11 491 293 493 458 427 395 328 301 299 266 298, 351 382 319 287 350 280 318 316 284 286 478 444 510 414 446 412, 457 41 489 43 459 425 393 361 363 331 265 329 297 267 330 296] 38 T ABLE XIV P RO P O S E D λ el F O R 1024 - Q A M . λ el [(195 707)) ((226 738 ) (1 9 3 705) (197 709) (229 741) (67 579) (225 737) (129 641), (163 675) (161 67 3) (165 677) (97 609) (69 581) (203 715) (65 577) (182 694), (54 566) (252 764) (222 734) (220 732) (18 0 692) (118 630) (176 688) (86 598), (151 663) (148 66 0) (150 662) (247 759) (246 758) (214 726) (212 724) (194 706), (199 711) (227 73 9) (231 743) (131 643) (133 645) (200 712) (99 611) (234 746), (202 714) (37 549) (101 613) (1 513) (33 545) (235 747) (201 713) (158 670), (156 668) (254 76 6) (22 534) (52 564) (470 982) (23 535) (20 532) (116 628 ), (183 695) (144 65 6) (146 658) (215 727) (242 754) (244 756) (213 725) (192 704), (228 740) (224 73 6) (162 674) (66 578) (64 576) (3 515) (5 517) (35 547), (451 963) (483 99 5) (139 651) (485 997) (481 993) (449 961) (233 745) (406 918), (190 702) (404 91 6) (407 919) (502 1014) (223 735) (48 560) (50 562) (468 980), (112 624) (178 69 0) (87 599) (240 752) (245 757) (210 722) (208 720) (196 708), (230 742) (130 64 2) (167 679) (205 717) (237 749) (450 962) (487 999) (453 965), (389 901) (385 89 7) (77 589) (72 584) (171 683) (137 649) (169 681) (438 950), (188 700) (94 606) (436 948) (400 912) (159 671) (191 703) (500 1012) (16 528), (255 767) (114 62 6) (179 691) (181 693) (84 596) (243 755) (198 710) (128 640), (98 610) (68 580) (232 744) (39 551) (448 960) (14 1 653) (138 650) (419 931), (355 867) (421 93 3) (74 586) (417 929) (75 587) (73 585) (126 638) (30 542), (124 636) (432 94 4) (434 946) (402 914) (92 604) (184 696) (496 1008 ) (18 530), (248 760) (119 63 1) (82 594) (147 659) (149 661) (134 646) (135 647) (103 615), (96 608) (239 751) (0 512) (136 648) (173 685) (10 4 616) (109 621) (13 5 25), (106 618) (323 83 5) (353 865) (107 619) (105 617) (62 574) (278 790) (28 54 0), (342 854) (374 88 6) (127 639) (95 607) (186 698) (471 983) (498 1010) (503 1015), (55 567) (221 733) (80 592) (145 657) (209 721) (132 644) (160 672) (71 583), (2 514) (455 967) (48 2 994) (170 682) (423 935) (359 871) (322 834) (45 557), (10 522) (11 523) (325 837) (357 869) (321 833) (60 572) (310 822) (304 816), (476 988) (31 543) (368 880) (370 882) (439 951) (403 915) (435 947) (152 664), (464 976) (253 76 5) (19 531) (115 627) (211 723) (166 678) (164 676) (7 519), (32 544) (480 992) (387 899) (168 680) (32 4 836) (320 832) (8 520) (456 968), (257 769) (461 97 3) (491 1003) (9 521) (41 553) (412 924) (510 1022 ) (306 8 18), (308 820) (274 78 6) (276 788) (372 884) (343 855) (336 848) (340 852) (189 701), (157 669) (51 563) (250 762) (177 689) (241 753) (70 582) (100 612) (4 51 6 ), (34 546) (452 964) (418 930) (327 839) (35 6 868) (495 1007) (490 1 002) (42 554), (261 773) (493 1005) (293 805) (43 5 55) (457 969) (446 958 ) (50 8 1020) (478 99 0 ), (272 784) (279 79 1) (307 819) (63 575) (56 568) (375 887) (371 883) (120 632), (154 666) (466 97 8) (218 730) (216 728) (83 595) (102 614) (204 716) (207 719), (484 996) (386 89 8) (76 588) (354 866) (260 772) (40 552) (291 803) (488 1000), (458 970) (289 80 1) (459 971) (393 905) (489 1001) (414 926) (444 956) (447 959), (408 920) (479 99 1) (511 1023) (275 787) (472 984) (58 570) (338 850) (122 634), (405 917) (499 1011) (469 981) (21 5 33) (117 629) (238 750 ) (36 548) (143 655), (175 687) (384 89 6) (79 591) (111 623) (47 559) (259 771) (295 807) (397 909), (429 941) (427 93 9) (363 875) (361 873) (425 937) (284 796) (286 798) (415 927), (440 952) (442 95 4) (410 922) (311 823) (504 1016) (506 1018) (24 536) (125 637), (401 913) (155 66 7) (467 979) (53 565) (85 597) (6 518) (486 998) (454 966), (420 932) (12 524) (15 527) (352 864) (292 804) (258 770) (392 904) (426 938), (394 906) (395 90 7) (301 813) (331 843) (329 841) (318 830) (380 892) (314 826), (348 860) (445 95 7) (413 925) (509 1021) (305 817) (61 573) (26 538) (433 945), (437 949) (187 69 9) (501 1013) (49 561) (113 625) (236 748) (140 652) (391 903), (416 928) (78 590) (326 838) (263 775) (256 768) (463 975) (431 943) (424 936), (365 877) (269 78 1) (328 840) (265 777) (297 809) (316 828) (280 792) (312 824), (376 888) (443 95 5) (411 923) (309 821) (477 989) (474 986) (273 785) (29 541), (93 605) (90 602) (88 600) (251 763) (17 529) (38 550) (206 718) (388 900), (108 620) (358 87 0) (294 806) (460 972) (290 802) (288 800) (399 911) (268 780), (367 879) (271 78 3) (360 872) (299 811) (267 779) (287 799) (350 862) (383 895), (315 827) (282 79 4) (378 890) (441 953) (475 987) (277 789) (507 1019) (27 539), (339 851) (123 63 5) (497 1009) (219 731) (81 593) (142 654) (172 6 84) (422 934), (14 526) (44 556) (494 1006) (492 10 0 4) (428 940) (270 78 2 ) (364 876) (33 2 844), (303 815) (264 77 6) (266 778) (330 842) (382 894) (319 831) (317 829) (283 795), (313 825) (379 89 1) (346 858) (409 921) (59 571) (373 885) (341 853) (369 881), (185 697) (153 66 5) (249 761) (217 729) (174 686) (390 902) (110 622) (46 558), (262 774) (396 90 8) (398 910) (430 942) (462 974) (302 814) (300 812) (333 845), (362 874) (298 81 0) (296 808) (351 863) (285 797) (381 893) (281 793) (347 859), (344 856) (377 88 9) (473 985) (505 1017) (25 537) (337 849) (121 633 ) (91 603), (465 977) (335 84 7) (366 878) (334 846) (349 861) (345 857) (57 569) (89 601)] 39 T ABLE XV P RO P O S E D λ ol F O R 1024 - Q A M . λ ol [(349 861) (334 846) (366 878) (345 857) (473 985) (398 910) (335 847) (505 1017), (57 56 9 ) (121 633) (91 6 03) (89 601) (465 977) (249 761) (217 729) (81 593), (113 625) (209 72 1 ) (208 720) (212 724) (195 707) (196 708) (198 710) (134 646), (102 614) (38 550) (142 654) (390 902) (422 934) (46 558) (396 908) (430 942), (377 889) (25 537) (337 849) (185 697) (153 665) (497 1009) (219 731) (49 561), (17 52 9 ) (85 597) (83 595) (211 723) (243 75 5 ) (210 722) (213 725) (214 726), (194 706) (199 71 1 ) (230 742) (135 647) (132 644) (166 678) (70 582) (6 518), (236 748) (140 65 2 ) (174 686) (110 622) (14 526) (262 7 7 4) (294 806) (302 814), (409 921) (341 85 3 ) (369 881) (123 635) (88 600) (501 1013) (53 565) (251 763), (117 629) (177 68 9 ) (145 657) (241 753) (245 757) (247 759) (242 754) (246 758), (226 738) (192 70 4 ) (228 740) (128 640) (160 672) (71 583) (164 676) (204 716), (238 750) (206 71 8 ) (388 900) (172 684) (108 620) (44 556) (494 1006) (428 940), (346 858) (373 88 5 ) (339 851) (93 6 05) (18 7 699) (499 1011) (467 979) (218 730), (216 728) (115 62 7 ) (181 693) (149 661) (240 752) (215 727) (244 756) (193 705), (227 739) (231 74 3 ) (224 736) (130 642) (162 674) (68 580) (100 612) (7 519), (36 54 8 ) (486 998) (454 966) (420 932) (78 590) (460 972) (492 1004) (462 974), (475 987) (59 571) (27 539) (433 945) (90 6 02) (15 5 667) (51 5 63) (21 533), (19 53 1 ) (82 594) (80 592) (84 596 ) (147 659) (146 658) (148 660) (150 662), (225 737) (197 70 9 ) (229 741) (131 643) (167 679) (103 615) (4 516) (32 544), (207 719) (484 99 6 ) (391 903) (416 928) (12 524) (326 8 3 8) (290 802) (270 782), (347 859) (507 1019) (29 541) (437 9 49) (40 1 9 1 3) (469 981) (466 978) (253 765), (250 762) (221 73 3 ) (179 691) (114 626) (144 656) (151 663) (86 598) (118 630), (67 57 9 ) (129 641) (133 645) (66 578) (98 61 0 ) (96 608) (239 7 51) (34 546), (455 967) (143 65 5 ) (175 687) (384 896) (358 870) (292 804) (288 800) (268 780), (379 891) (277 78 9 ) (273 785) (125 637) (122 634) (405 917) (157 669) (464 976), (55 56 7 ) (119 631) (112 624) (87 599) (183 695) (176 688) (180 692) (182 694), (161 673) (163 67 5 ) (99 611) (64 576) (3 515) (205 717) (232 7 44) (2 514), (452 964) (386 89 8 ) (418 930) (76 5 88) (15 527) (263 775) (258 770) (399 911), (344 856) (309 82 1 ) (61 573) (26 538) (371 883) (120 632) (154 666) (503 1015), (255 767) (248 76 0 ) (16 528) (468 9 80) (17 8 690) (116 628) (220 732) (22 534 ), (65 57 7 ) (165 677) (200 712) (234 746) (5 51 7) (237 749) (39 551) (0 512 ), (480 992) (168 68 0 ) (327 839) (79 5 91) (35 4 866) (256 768) (463 975) (364 876), (411 923) (441 95 3 ) (305 817) (506 1018) (24 536) (338 850) (189 701) (152 664), (496 1008) (498 1010) (18 530) (23 5 3 5) (20 532) (52 564) (222 734) (203 7 1 5), (97 60 9 ) (69 581) (101 6 13) (20 2 714) (487 999) (450 962) (448 960) (482 994), (170 682) (423 93 5 ) (111 623) (260 772) (352 864) (47 559) (431 943) (300 812), (313 825) (443 95 5 ) (477 989) (474 986) (58 570) (375 8 8 7) (340 852) (435 947), (471 983) (184 69 6 ) (50 562) (223 7 35) (50 2 1014) (254 766) (252 764) (54 566), (201 713) (1 513) (37 549) (483 995) (35 547) (138 650) (141 653) (136 648), (387 899) (324 83 6 ) (356 868) (40 5 52) (25 9 771) (392 904) (424 936) (332 844), (283 795) (376 88 8 ) (410 922) (504 1016) (472 984) (56 568 ) (336 848) (403 915), (439 951) (186 69 8 ) (500 1012) (191 703) (48 560) (470 982 ) (158 670) (156 668), (235 747) (33 545) (485 997) (451 963) (453 965) (389 901) (173 685) (104 61 6 ), (359 871) (320 83 2 ) (495 1007) (490 10 0 2) (291 803) (426 938) (367 879) (271 783 ), (381 893) (378 89 0 ) (413 925) (509 1021) (275 787) (343 855) (372 88 4 ) (370 882), (127 639) (92 604) (402 914) (159 671) (407 919) (400 912) (404 916) (233 74 5 ), (481 993) (406 91 8 ) (190 702) (139 651) (385 897) (419 931) (109 621) (355 867), (322 834) (8 520) (456 9 68) (29 5 807) (488 1000) (365 877) (269 781) (333 845), (317 829) (281 79 3 ) (282 794) (445 957) (311 823) (307 819) (63 575) (276 788), (95 60 7 ) (368 880) (432 944) (434 946) (94 606) (124 636) (436 948) (44 9 961), (137 649) (438 95 0 ) (171 683) (417 929) (72 584) (421 9 3 3) (106 618) (13 525), (45 55 7 ) (261 773) (42 5 54) (39 7 909) (394 906) (301 813) (303 815) (264 776), (383 895) (315 82 7 ) (314 826) (348 860) (442 954) (408 920) (511 1023) (274 786), (308 820) (31 543) (374 886) (342 854) (28 540 ) (30 542) (73 585) (188 700), (169 681) (75 587) (107 619) (74 586) (323 835) (77 589) (10 522) (257 769), (493 1005) (461 973) (429 941) (427 939 ) (395 907) (265 777) (360 872) (298 810), (285 797) (319 83 1 ) (280 792) (312 824) (286 798) (440 952) (479 991) (279 791), (272 784) (304 81 6 ) (476 988) (306 818) (310 822) (60 572) (105 617) (62 5 74), (278 790) (126 63 8 ) (353 865) (357 869) (11 523) (325 8 3 7) (491 1003) (293 805), (289 801) (458 97 0 ) (393 905) (331 843) (328 840) (299 811) (362 874) (296 808), (351 863) (382 89 4 ) (287 799) (350 862) (380 892) (318 830) (284 796) (415 927), (447 959) (478 99 0 ) (444 956) (508 1020) (457 969) (510 1022) (41 553) (9 521), (321 833) (412 92 4 ) (489 1001) (446 958) (414 926) (43 555 ) (425 937) (361 873), (459 971) (363 87 5 ) (329 841) (297 809) (316 828) (267 779) (266 778) (330 842)] 40 A C K N OW L E D G M E N T R E F E R E N C E S [1] G. 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Novel Method for Multi-Dimensional Mapping of Higher Order Modulations for BICM-ID Over Rayleigh Fading Channels

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