Lower Bounds on the Minimum Pseudodistance for Linear Codes with $q$-ary PSK Modulation over AWGN

Lo wer Bounds on the Minimum Ps eudodistance for Linear Codes with q -ary PSK Modu lati on o v er A WGN V italy Skache k Claude Shannon Institute, Un i versity College Dub lin Belfield, Dublin 4, I reland Email:vitaly .skache k@ucd.ie Mark F . Flanagan DEIS, Un i versity of Bologna V ia V enezia 52, 47023 Cesena, Italy Email: mark.flanagan@ieee.o rg Abstract —W e present lo wer b ounds on the minimum pseu- docodeword effective Eu clidean d istance (or minimum “pseu- dodistance”) fo r coded modulation systems using linear codes with q -ary phase-shift keying (PS K) modulation over the ad- ditive whit e Gaussian noise (A WGN) channel. These bou nds apply to both binary and nonb inary co ded modulation systems which use di rect modulation mapping of coded symbols. The minimum pseudodistance ma y serv e as a first-order measur e of error -correc ting perfo rmance fo r both linear-pr ogramming and message-passing b ased recei vers. In the case of a linear - program ming b ased rec eiv er , the mini mum pseudodistance may be used to form an exact boun d on the codewor d err or rate of the system. K eywords: Iterativ e decoding, linear-pr ogramming decoding, factor graph, gr aph cover , pseudocodewords, pseud odistance. I . I N T RO D U C T I O N A. Backgr ound In classical codin g th eory , max imum-likelihoo d (ML) de - coding of a sign al-space code le ads to a nea r est-neig hbour decision rule in the signal space. For this reason, the minimum Euclidean distance between mod ulated codew ords (signal points) of a signal-space cod e is u sed as a first-o rder measure of its er ror-correcting perfor mance un der ML d ecoding. In the case of bin ary modulation, the minim um Hamming distance of the under lying co de may be substituted, since in this case the Hamming d istance is pro portional to the squared E uclidean distance. Recently , lo w-d ensity p arity check ( LDPC) codes [1] h a ve attracted much interest du e to their p ractical efficiency . In particular, it was shown that several families of LDPC codes can attain the capacity of various chann els, when deco ded by itera ti ve m essage pa ssing (MP) algorithms (for in stance see [ 2], [3], [4]). The MP decoding algorithm o perates locally on the T ann er graph , a grap h which repr esents the parity-ch eck matrix. The notion o f comp utation tr ee pseudoc odewor ds was introduc ed in [5] in order to adequately explain the limitations of MP de- coding of binary LDPC codes. Compu tation tree pseudo code- words ar e closely related to graph-cover pseudo codewor ds . The latter w ere extensi vely studied in [ 6], [ 7], [ 8] an d [9 ]. The gr aph-cover pseud ocodewords lie inside a region called the funda mental co ne (see [ 6], [7]). Th e set of g raph-cover pseudoco dew o rds were shown to be e quiv alent to the set of linear-pr ogramming ( LP) pseudoco dewor ds for the cases of binary [10], [11] and non binary coded mod ulation sys- tems [12], [13]. In b oth b inary and nonbina ry cases, necessary and sufficient c onditions for codew ord error under linear progr amming (LP) dec oding cou ld b e expressed is terms of these LP pseudocodewords, assuming transmission o f the all- zero c odew ord ([11], [12]). In [1 4], the p seudocod e word effecti ve Euclide an distance, or pseudodistance , is associated with any pseudo codeword. This con cept of pseu dodistance was sho wn in [14] to play an ana lagous role to tha t of th e signal Euc lidean distance in ML decoding . T he minimum pseudodistance is defined as the minimum over all pseudod istances of pseudo codewords; this may be taken as a first-o rder measure of decod er erro r- correcting per formance for L P or MP decod ing. In [9], it was shown that bo unds co uld be obtained on th e minimum pseudoweight of a binar y linear code, these b ounds bein g expressed in terms of param eters of the parity -check matrix of th e co de. In this work, we e x tend th e results in [9] to the non binary case. In particular, we show that bounds on the minimu m pseudod istance can be obtaine d for the case of n onbinar y coding an d modu lation, which ar e gener alizations of the bound s on pseudoweight in [9] fo r th e case o f b inary c oding and mo dulation. Generally , the tech niques are based on the technique s therein, althou gh some a dditional ideas are used. B. Basic Definitio ns W e consider codes over finite rings (this includes codes over finite fields, but ma y be mor e g eneral). Denote by R a ring with q elemen ts, by 0 its ad diti ve identity , and let R − = R \{ 0 } . Let C be a linear [ n, k ] code with parity-check matrix H over R (we assume that R is quasi-Frobeniu s, which implies that the parity-check matrix exists). Th e p arity che ck matrix H has m ≥ n − k r ows . Denote the set of colum n indices an d the set of row indices o f H by I = { 1 , 2 , · · · , n } and J = { 1 , 2 , · · · , m } , respectively . W e use no tation H j for the j -th row of H , and I j for th e sup port of H j . Giv en an y c ∈ R n , we say that pa rity check j ∈ J is satisfied by c if a nd on ly if X i ∈I j c i · H j,i = 0 . (1) Also, we say that th e vector c is a cod e word of C , writing c ∈ C , if an d o nly if all parity checks j ∈ J a re satisfied b y c . Let the grap h G = ( V , E ) be the T anner graph of C associated with the parity -check matrix H . This graph has vertex set V = { u 1 , u 2 , · · · , u n } ∪ { v 1 , v 2 , · · · , v m } , a nd th ere is an edge b etween u i and v j if an d only if H j,i 6 = 0 . This e dge is labelled with th e value H j,i . W e denote by N ( v ) the set o f n eighbor s of a vertex v ∈ V . For a word c = ( c 1 , c 2 , · · · , c n ) ∈ R n , we associate the value c i with variable verte x u i for each i ∈ I . It may be ea sily seen that the T anne r graph provides a g raphical means o f check ing wh ether each p arity-check j ∈ J is satisfied, and hen ce whethe r the vector c is a code word of C . Definition 1 .1: ([6]) A g raph ˜ G = ( ˜ V , ˜ E ) is a finite c over of the grap h G = ( V , E ) if there exists a map ping Π : ˜ V → V which is a g raph hom omorph ism ( Π takes adjacen t vertice s of ˜ G to adjacent vertices of G ), su ch that fo r ev er y vertex v ∈ G and e very ˜ v ∈ Π − 1 ( v ) , the neighborh ood N (˜ v ) of ˜ v (inc luding edge labels) is mapped bijectively to N ( v ) . Definition 1 .2: ([6]) A cover of the grap h G is c alled an M -c over , wh ere M is a positi ve integer , if | Π − 1 ( v ) | = M f or ev e ry vertex v ∈ V . Fix some positi ve integer M . Let ˜ G = ( ˜ V , ˜ E ) be an M -c over o f th e T an ner gr aph G = ( V , E ) of the code C associated with the parity-c heck matrix H . Den ote the vertices in the sets Π − 1 ( u i ) and Π − 1 ( v j ) by { u i, 1 , u i, 2 , · · · , u i,M } and { v j, 1 , v j, 2 , · · · , v j,M } , resp ecti vely , where i ∈ I an d j ∈ J . Consider the line ar code ˜ C of length M n over R , defined by the M m × M n parity-check matrix ˜ H . For 1 ≤ i ∗ , j ∗ ≤ M and i ∈ I , j ∈ J , we let i ′ = ( i − 1) M + i ∗ , j ′ = ( j − 1 ) M + j ∗ , and ˜ H j ′ ,i ′ =  H j,i if u i,i ∗ ∈ N ( v j,j ∗ ) 0 otherwise . (2) Then, any vector p ∈ ˜ C has the form p = ( p 1 , 1 , p 1 , 2 , · · · , p 1 ,M , p 2 , 1 , p 2 , 2 , · · · , p 2 ,M , · · · , p n, 1 , p n, 2 , · · · , p n,M ) . W e associate th e value p i,ℓ ∈ R with th e vertex u i,ℓ in ˜ G ( i ∈ I , ℓ = 1 , 2 , · · · , M ). I t may be seen that ˜ G is the T anner graph of the cod e ˜ C associated with the parity-ch eck matrix ˜ H . The word p ∈ ˜ C as above is called a graph-cover pseudoco dewor d of the code C . W e also define the n × q pseudoco dewor d matrix corr esponding to p by P =  m ( α ) i  i ∈I ; α ∈ R , where m ( α ) i = |{ ℓ ∈ { 1 , 2 , · · · , M } : p i,ℓ = α } | ≥ 0 , for i ∈ I , α ∈ R . W e th en defin e th e n ormalized pseudoc ode- wor d matrix co rrespond ing to p b y P 0 =  f ( α ) i  i ∈I ; α ∈ R , where f ( α ) i = m ( α ) i / M for every i ∈ I , α ∈ R . In [12], [15], anoth er set of pseudoc odew ords, called linea r - pr ogramming pseudocodewor d s , was defined . Th ese LP pseu- docod e word s, wh ich also a dmit a matrix representation, were shown to be directly linked to codew o rd error e vents in LP decod ing. It w as also shown in [1 2], [15] that the two pseudoco dew o rd concepts ar e equivalent, i.e. th ere exists an LP pseudocodeword with a particular pseu docodeword ma trix if an d o nly if the re exists a graph-cover pseu docodeword having the same pseudocod ew or d m atrix. It was shown in [12] and [ 16] that fo r the ca se of q - ary PSK transmission over A WGN u nder LP or MP decodin g, the code word error rate perf ormance is independ ent of the transmitted cod e word under th e following condition s. First, R under addition is a cyclic gro up. I f we let β be a ge nerator in R then we may write R = { 0 , β , 2 β , · · · , ( q − 1) β } where k β = β + · · · + β ( k > 0 term s in s um). Second, the modulation mapping is the ‘na tural’ mapp ing M ( k β ) = exp  ı · 2 π k q  , (3) where ı = √ − 1 . W e assume in this work th at these con ditions hold; hen ce in th e sequel, we adopt the simp ler notation f i ( k ) for f ( kβ ) i , k = 0 , 1 , · · · , q − 1 . I I . B O U N D S O N T H E P S E U D O D I S TA N C E O F I N D I V I D U A L P S E U D O C O D E W O R D S In [11], for the case of binary coding and binary modulation, the set of pseudocodewords was used to cha racterize the error correction capability of the system und er LP deco ding. This was extended to the case of n onbinary coding and mod ulation in [1 2]. In [1 4], it was ob served that with eac h pseudoco de- word p may be associated a p oint in the signal space ; these signal p oints the n play a role in LP decod ing analagou s to that of the mo dulated codewords in ML decodin g. In particular, we may associate with each pseudo codeword an effecti ve Euclidean d istance f rom th e m odulated all-ze ro cod ew or d, or pseudod istance, d eff ( p ) (pseudo distance with respect to the all- zero codeword is su f ficient assumin g the symm etry conditio n above). Then, assuming LP d ecoding, the event E ( p ) = “on transmission o f the a ll-zero co dew o rd, th ere is a codeword error du e to pseudocod e word p ” has pr obability P ( E ( p )) = Q  d eff ( p ) 2 σ  (4) (where σ 2 is the n oise variance per dim ension, and Q ( x ) = 1 2 π R ∞ x exp( − t 2 / 2) dt d enotes the Gaussian Q -fun ction) an d thus the probability of co dew o rd err or is eq ual to P ( S E ( p )) where the un ion is over the set of all pseudoco dew o rds p (equation (4) was st ated in [14] for the case of MP de coding and comp utation tr ee pseudo codewords). Therefore the min- imum pseudodistance d min = min p { d eff ( p ) } may b e taken as a first-ord er m easure of error-correcting per formance of the cod ed modulatio n system. For the case o f MP d ecoding and grap h-cover pseudoc odew ords, (4) may be taken as an approx imation. Also, for the case of binary coding and mod- ulation, the pseudocod ew or d e ffecti ve Hamming weig ht (or “pseudoweight”) may be defined by w eff ( p ) = d 2 eff ( p ) / 4 by analogy with the case of classical ML dec oding [1 4]. It was shown in [1 7] tha t for the case of q -a ry PSK modulatio n over A WGN, the square d pseud odistance between the all-zero codeword and the pseudocodeword p is given by d 2 q ( p ) = S 2 V , (5) where S = 2 X i ∈I 1 − q − 1 X k =0 f i ( k ) · cos  2 π k q  ! , (6) and V = X i ∈I q − 1 X k =0 f 2 i ( k ) + 2 X k<ℓ k,ℓ ∈{ 0 ,. . .,q − 1 } f i ( k ) f i ( ℓ ) · cos  2 π ( k − ℓ ) q  − 2 q − 1 X k =0 f i ( k ) · cos  2 π k q  + 1 ! . (7) By r earrangem ent of the e x pression in (6), we ha ve S = 2 X i ∈I 1 − q − 1 X k =0 f i ( k ) · cos  2 π k q  ! = 2 X i ∈I q − 1 X k =1 f i ( k ) ·  1 − cos  2 π k q  ! ≥ 2  1 − cos  2 π q  · X i ∈I q − 1 X k =1 f i ( k ) ! . (8) Similarly , for (7) we ha ve V = X i ∈I q − 1 X k =1 f 2 i ( k ) + f 2 i (0) + 2 X k<ℓ k,ℓ ∈{ 1 ,. . .,q − 1 } f i ( k ) f i ( ℓ ) · cos  2 π ( k − ℓ ) q  + 2 q − 1 X ℓ =1 f i (0) f i ( ℓ ) · cos  2 π ℓ q  − 2 q − 1 X k =1 f i ( k ) · cos  2 π k q  − 2 f i (0) + 1 ! . It follo ws that V ≤ X i ∈I q − 1 X k =1 f i ( k ) ! 2 + (1 − f i (0)) 2 + 2 q − 1 X k =1 f i ( k ) cos  2 π k q  · ( f i (0) − 1) ! . After re arrangem ent, w e ob tain V ≤ 4 X i ∈I q − 1 X k =1 f i ( k ) ! 2 . (9) W e substitute the expressions in (8) and (9) into (5), and obtain that d 2 q ( p ) ≥ (1 − cos(2 π / q )) 2 ·  P i ∈I x i  2 P i ∈I x 2 i , (10) where ∀ i ∈ I : x i = q − 1 X k =1 f i ( k ) . Example 2.1 : T ake R = { 0 , 1 } with binar y signaling over A WGN. In this case, q = 2 , and (10) ca n be re- written as d 2 2 ( p ) ≥ 4  P i ∈I x i  2 P i ∈I x 2 i , which accords with the well-known pseudoweight expression for th e case of binary code and modulation [6]. Example 2.2 : T ake R = Z 3 = { 0 , 1 , 2 } with ternary PSK modulation over A WGN. W e show that in this case the inequality ( 10) can be slightly improved. Observe that in this case, (6) and (7) can be re-written as S = 2 X i ∈I  1 2 f i (1) + 1 2 f i (2) + (1 − f i (0))  = 3 X i ∈I ( f i (1) + f i (2)) , (11) and V = X i ∈I  3( f i (1)) 2 + 3( f i (2)) 2 + 3 f i (1) f i (2)  ≤ 3 X i ∈I  ( f i (1)) + ( f i (2))  2 , (12) where the last equality in (11) an d the equ ality in (12) are due to the fact that P q − 1 k =0 f i ( k ) = 1 for all i ∈ I . Finally , we substitute th e expressions in (11) and (1 2) into (5) to o btain that d 2 3 ( p ) ≥ 3 ·  P i ∈I x i  2 P i ∈I x 2 i . (13) Example 2.3 : T ake R = Z 4 = { 0 , 1 , 2 , 3 } with quater- nary PSK (QPSK) modu lation over A WGN chann el. In this case, using (10), we obtain that d 2 4 ( p ) ≥  P i ∈I x i  2 P i ∈I x 2 i , (14) where x = ( x i ) n i =1 = ( f i (1) + f i (2) + f i (3)) n i =1 . I I I . I N E Q U A L I T I E S F O R P S E U D O C O D E W O R D S A complete char acterization of the fundamental co ne , in which the pseudocodewords lie, was gi ven for th e case of binary coding and mod ulation in [6]. For the presen t m ore general fr amew o rk, a complete ch aracterization of the c orre- sponding fun damental r egion app ears to be a difficult task. In this section we derive a set o f inequalities which must be satisfied by th e entries of a ny pseudocod ew or d matrix ; these inequalities m ust n ecessarily be satisfied by any pseudocod e- word lying in the fu ndamental con e. These inequalities will b e helpful in deri v ing the b ounds on minimum pseudodistan ce in the sequel. Theor e m 3.1: Let C be a linear [ n, k ] code over R with an m × n par ity-check matrix H . Let I , J and I j be defined as in Section I-B. Assume that H j,i is not a zero-divisor in R for a ny j ∈ J , i ∈ I j . Let P =  m ( α ) i  i ∈I ; α ∈ R be the p seudocod e word matrix of a gra ph-cover pseudocod e- word p of the co de C with parity- check m atrix H . Then, for any j ∈ J , ℓ ∈ I j ) , X i ∈I j \{ ℓ } X b ∈ R − m ( b ) i ≥ X b ∈ R − m ( b ) ℓ . (15) Pr oof: Sup pose the graph- cover pseudocod e word p cor- respond s to the M -cover ˜ G = ( ˜ V , ˜ E ) , and let ˜ C be the linear code of length M n ov er R defined by the parity-check m atrix ˜ H described by (2). Then, ˜ G is the T anner g raph o f the code ˜ C associated with the parity-check matrix ˜ H . T ake some j ∈ J and ℓ ∈ I j . Fix som e 1 ≤ j ∗ ≤ M , and take the j ∗ -th co py v j,j ∗ ∈ ˜ V of the par ity-check vertex v j ∈ V . Let  u i,σ ( i,j ∗ )  i ∈I j = N ( v j,j ∗ ) ⊆ ˜ V , where 1 ≤ σ ( i, j ∗ ) ≤ M for every i ∈ I j . Denote j ′ = ( j − 1) M + j ∗ . Since p ∈ ˜ C , X i ∈I j H j ′ , ( i − 1) M + σ ( i,j ∗ ) · p i,σ ( i,j ∗ ) = 0 . This can be rewritten as X i ∈I j H j,i · p i,σ ( i,j ∗ ) = 0 . (16) Assume that p ℓ,σ ( ℓ,j ∗ ) 6 = 0 . Th en, X i ∈I j \{ ℓ } H j,i · p i,σ ( i,j ∗ ) = − H j,ℓ · p ℓ,σ ( ℓ,j ∗ ) , (17 ) and, since H j,ℓ is not a zero divisor in R , the expression in (17) is non-zero. Th erefore, there exists at least one i j ∗ ∈ I j , i j ∗ 6 = ℓ , suc h th at p i j ∗ ,σ ( i j ∗ ,j ∗ ) 6 = 0 . The num ber of ind ices j ∗ ( 1 ≤ j ∗ ≤ M ) su ch that p ℓ,σ ( ℓ,j ∗ ) 6 = 0 is gi ven by P b ∈ R − m ( b ) ℓ . This numb er is equal to th e numb er of indices j ∗ ( 1 ≤ j ∗ ≤ M ) su ch that p i j ∗ ,σ ( i j ∗ ,j ∗ ) 6 = 0 , wh ich, in turn, is less than or equal to X i ∈I j \{ ℓ } X b ∈ R − m ( b ) i . On di vision of both sides of (15) by M , we obtain the following result. Cor ollary 3.2: L et C , H and P be defined as in Theo - rem 3 .1. Th en, for any j ∈ J , ℓ ∈ I j , X i ∈I j \{ ℓ } q − 1 X k =1 f i ( k ) ≥ q − 1 X k =1 f ℓ ( k ) . (18) I V . E I G E N V A L U E B O U N D In this section, we consider ( c, d ) -regular codes, i.e. the parity check matrix H of C has c no nzero elements per colu mn and d non zero elements per ro w . T hrough out this section , let C be a ( c, d ) -regular linear [ n, k ] code over R with an m × n parity-ch eck matrix H , and assume that H j,i is not a z ero- divisor in R for any j ∈ J , i ∈ I j . Let P 0 = ( f i ( k )) i ∈I ; k ∈{ 1 , 2 , ··· , q − 1 } be the no rmalized pseudo codeword matrix of a graph -cover pseudoco dew o rd p corr esponding to some M - cover of the T anne r graph of H . Denote ∀ i ∈ I : x i = q − 1 X k =1 f i ( k ) and x = ( x i ) i ∈I . W e define a real matr ix L = H T s · H s , where H s is an m × n matr ix whose entries are equ al to one on the suppor t of H , and are equal to zer o o therwise. W e assume th at the T anne r graph of C corresp onding to H consists of a single connected com ponent, and denote b y λ 1 ≥ λ 2 ≥ · · · ≥ λ n the eigenv alues of L . Let v 1 , v 2 , · · · , v n be the set o f orthonor mal eigenv e ctors correspon ding to eigenv alu es λ 1 , λ 2 , · · · , λ n of the matrix L , respectively . Then , λ 1 = c · d > λ 2 , and v 1 = 1 √ n · 1 . Also, assume that q -ary PSK m odulation is u sed for transmission over the A WGN chan nel. Our an alysis fo llows the lines of [9]. Lemma 4. 1: Let P 0 and x be defined as above. Then, for any j ∈ J , we have   X i ∈I j x i   2 ≥ 2 · X i ∈I j x 2 i . Pr oof: F o r any j ∈ J write   X i ∈I j x i   2 =   X i ∈I j x i   ·   X ℓ ∈I j x ℓ   = X i ∈I j x i   X ℓ ∈I j x ℓ   ≥ X i ∈I j 2 · x 2 i , where th e inequ ality is due to Corollary 3.2. The fo llowing lemm a is the cou nterpart of L emma 8 in [ 9]. Lemma 4.2: Let x be a vector defined a s in Lemma 4. 1, and let y = x · H T s . Then , || y || 2 2 ≥ 2 c · || x || 2 2 . Pr oof: W e wr ite || y || 2 2 = X j ∈J y 2 j = X j ∈J   X i ∈I j x i   2 . W e apply Lemma 4.1 to obtain that || y || 2 2 ≥ X j ∈J 2 · X i ∈I j x 2 i = 2 c · || x || 2 2 , where the last transition is due to the fact that each colum n in H con tains exactly c nonzero symbols. The f ollowing lemma is based on Theorem 10 in [9]. Lemma 4.3: Let x and y be vectors defined as in Lemma 4.2, and let λ 1 and λ 2 be defined as in Section IV. Then, || y || 2 2 ≤ λ 1 − λ 2 n X i ∈I x i ! 2 + λ 2 || x || 2 2 . Pr oof: Write x as x = n X i =1 σ i v i , where v i ( i ∈ I ) are defined in Section I V, and σ i ( i ∈ I ) are rea l nu mbers. In particular , σ 1 = 1 √ n h x , 1 i = 1 √ n n X i =1 x i ! . Then, || y || 2 2 = || x · H T s || 2 2 = x · L · x T = n X i =1 σ i v i · L · n X i ′ =1 σ i ′ v i ′ = n X i =1 σ i v i · n X i ′ =1 λ i ′ σ i ′ v i ′ = n X i =1 λ i σ 2 i = λ 1 n n X i =1 x i ! 2 + n X i =2 λ i σ 2 i ≤ λ 1 n n X i =1 x i ! 2 + λ 2 n X i =2 σ 2 i = λ 1 n n X i =1 x i ! 2 + λ 2 n X i =1 σ 2 i − σ 2 1 ! = λ 1 n n X i =1 x i ! 2 + λ 2 n X i =1 σ 2 i || v i || 2 2 ! − λ 2 σ 2 1 = λ 1 n n X i =1 x i ! 2 + λ 2 || x || 2 2 − λ 2 n n X i =1 x i ! 2 , as claim ed. The following theorem summarizes the ma in result in this section. Theor e m 4.4: Let C , H , H s and L b e d efined as above. Then the minimum pseudodistance with q -a ry PSK modulation over the A WGN channel is bounded fro m below by d 2 min ,q ≥ (1 − cos(2 π /q )) 2 · n · 2 c − λ 2 λ 1 − λ 2 . Pr oof: The c ombination o f th e resu lts in Lemmas 4.2 and 4 .3 imm ediately leads to λ 1 − λ 2 n X i ∈I x i ! 2 + λ 2 || x || 2 2 ≥ 2 c · || x || 2 2 , and b y rear rangemen t of the co efficients we ob tain  P i ∈I x i  2 || x || 2 2 ≥ n · 2 c − λ 2 λ 1 − λ 2 . (19) By r e-writing (10), we get d 2 min ,q ≥ (1 − cos(2 π /q )) 2 ·  P i ∈I x i  2 || x || 2 2 ≥ (1 − cos(2 π /q )) 2 · n 2 c − λ 2 λ 1 − λ 2 , (20) where th e last transition is due to (19). Example 4.1 : Co nsider R = { 0 , 1 } with b inary signaling over A WGN. In that case, q = 2 , and so (20) can be re-written as d 2 min , 2 ≥ 4 n · 2 c − λ 2 λ 1 − λ 2 , which co incides with the corr esponding bound in [9], since in this case d 2 eff , 2 ( p ) / 4 = w eff ( p ) , the ef fective Hamming weight of th e pseu docodeword p . Example 4.2 : T ake R = Z 3 with ternary PSK over A WGN, as in Example 2.2. I n th is case, we can combine ( 13) with (19), thus ob taining d 2 min , 3 ≥ 3 ·  P i ∈I x i  2 P i ∈I x 2 i ≥ 3 n · 2 c − λ 2 λ 1 − λ 2 . Note that this boun d is better then the respecti ve bo und which follows directly f rom (20). Example 4.3: T ake R = Z 4 with QPSK over A WGN, a s in Exa mple 2.2. In this case, we can combine (14) with (19), thus ob taining d 2 min , 4 ≥  P i ∈I x i  2 P i ∈I x 2 i ≥ n · 2 c − λ 2 λ 1 − λ 2 . V . L I N E A R - P RO G R A M M I N G B O U N D In this section, we p resent the linear-program ming lower bound on the m inimum p seudodistance , similar to its cou n- terpart in [9, Sec tion 4]. Throug hout this section, let C be a linear [ n, k ] code over R with an m × n parity-ch eck matrix H , and a ssume that H j,i is no t a zero-divisor in R for a ny j ∈ J , i ∈ I j . Let P 0 = ( f i ( k )) i ∈I ; k ∈{ 1 , 2 , ··· , q − 1 } be the no rmalized pseudo codeword matrix of a gr aph-cover pseudoco dew o rd p c orrespon ding to some M -cover of the T anne r grap h of H . Denote ∀ i ∈ I : x i = q − 1 X k =1 f i ( k ) . It follo ws from Corollary 3.2 , th at X i ∈I j \{ ℓ } x i ≥ x ℓ (21) for all j ∈ J , ℓ ∈ I j . The inequa lities (21) (for all j ∈ J , ℓ ∈ I j ) can be e x pressed as K x T ≥ 0 , (22) for so me K . Let the entr ies o f a vecto r y ∈ R ( I 2 ) be indexed by ( i, i ′ ) ∈ I 2 . For i ∈ I we denote by y ( i, :) and y (: ,i ) the sub-vectors of length n of y co nsisting of all entries y ( i,i ′ ) for all i ′ ∈ I and o f all entries y ( i ′ ,i ) for all i ′ ∈ I , respectively . The f ollowing th eorem is th e main result of this section . It is a generalization o f The orem 15 in [9]. Theor e m 5.1: For q -ary PSK m odulation over A WGN, the minimum pseudod istance d min ,q is boun ded from b elow by d 2 min ,q ≥ ( 1 − cos(2 π /q )) 2 · 1 max y ∈K 1 { f ′ ( y ) } , where f ′ ( y ) = X i ∈I y ( i,i ) , and K 1 =    y ∈ R ( I 2 )       y ≥ 0 , y · 1 T = 1 , K y T ( i, :) ≥ 0 T for all i ∈ I , K y T (: ,i ) ≥ 0 T for all i ∈ I    . Sketch of the p r oof: W e start w ith the expr ession in (10). The expr ession  P i ∈I x i  2 P i ∈I x 2 i can be bounded from below using the same techniq ues as in the proof of The orem 15 in [9], with re spect to x defined as above. W e o mit the details. A C K N O W L E D G E M E N T S The authors would like to than k E. Byrne, M. Gre ferath and D. Sridh ara for helpful d iscussions. This work was supp orted in part by the Claud e Shann on In stitute for Discrete Math e- matics, Coding and Cryptog raphy (Science Foundation Ireland Grant 0 6/MI/006) . R E F E R E N C E S [1] R. G. 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