A Construction for Constant-Composition Codes

1 A Construction for Constant-Composition Codes Y ang Ding Abstract — By employing the residue polynomials, we gi ve a construction of constant-composition codes. This construction generalizes the one proposed by Xing [16]. It turns out that when d = 3 this construction g iv es a lowe r bound of constant- composition codes improving the one in [10] for some case. Moreo ver , fo r d > 3 , we giv e a lower bound on maximal size of constant-composition codes. In particular , our bound for d = 5 giv es the best p ossible size of constant-composition codes up to magnitude. Index T erms — constant-composition codes, g enus, re sidue poly- nomial, rational function field s. I . I N T RO D U C T I O N Constant-com position codes are a sub class of constant weight codes, in which both weigh t restrict and element composition r estrict ar e inv olved. T he class o f constant- composition codes h av e attr acted recent intere st due to its numero us application s, such as in determin ing th e zero erro r decision feedback capacity of discrete memoryless channels [15], multip le-access communications [8], spherical codes for modulatio n [9], DN A codes [11], powerline commu nications [2], and frequen cy h opping [3 ]. One of the most fundam ental prob lem in codin g theory is the problem o f determining th e m aximum size of a bloc k code, giv en its length and minimum distance. The p roblem of determinin g the maximum size of a constan t-compo sition code is much less understoo d than the con stant-weight an d linea r cases. In the recent year s, research es consider the problems of maximizing the size of a co nstant-com position code ( see [1], [10], [1 3]), and constructin g op timal code s to achieve these bound s (see [4], [5], [6], [7], [14]). In this pap er , we give a con struction for con stant-compo sition codes then pro duce a lower bound on constant-co mposition codes for arbitrary gi ven minimum distance. W e show that when q = 3 and d = 5 , our bound gives the best possible size o f con stant-comp osition codes up to magnitud e. As far a s we know , excep t for the bound giv en in this paper, th ere is no boun ds on d > 3 so far . This correspon dence is o rganized as follows. In Section II, we introdu ce so me basic d efinitions and no tations. W e also r evie w some ba sic pr operties which will be used in th is correspo ndence. The main constructio n is p resented in Section III. In Sec tion IV , Theore m 1 in section II are used to ob tain some g ood lower b ounds o n co nstant-com position codes. This work wa s supported by the China Scholarship Council. The author is with the Departmen t of Mathematics, Southeast Uni ver sity , Nanjing, 210096, People’ s Republ ic of China (e-mail: PG23067461@nt u.edu.sg). This work was carrie d out while the author was studying in Division of Mathema tical Sciences, Schoo l of Physical and Mathemat ical Sciences, Nan yang T echnologic al Unive rsity , Singapore under the exc hange program. I I . P R E L I M I NA RY W e use the stand ard notations for cod es as follows. Let Z q denote the set { 0 , 1 , · · · , q − 1 } , and let Z n q be the set o f all n -tup les over Z q , where q is a positiv e inte- ger . Let V n, [ ω 0 , ω 1 , ··· , ω q − 1 ] ( q ) denote the set of n -tup les over Z q of the fixed comp osition [ ω 0 , ω 1 , · · · , ω q − 1 ] , i.e., the number of 0 ’ s, 1 ’ s, · · · , q − 1 ’ s in the n - tuple over Z q is given by ω 0 , ω 1 , · · · , ω q − 1 , respec ti vely , where n = ω 0 + ω 1 + · · · + ω q − 1 . It is o bvious th at V n, [ ω 0 , ω 1 , ··· , ω q − 1 ] ( q ) c ontains  n ω 0 , ω 1 , · · · , ω q − 1  ele- ments. An ( n, M , d, [ ω 0 , ω 1 , · · · , ω q − 1 ]) q constant- composition code C is a subset of V n, [ ω 0 , ω 1 , ··· , ω q − 1 ] ( q ) with size M and m inimum Hamming distance d . W e use A q ( n, d, [ ω 0 , ω 1 , · · · , ω q − 1 ]) to deno te the max imum size of an ( n, M , d, [ ω 0 , ω 1 , · · · , ω q − 1 ]) q constant-co mposition code. In o rder to establish our results in th is co rrespon dence, we need the fo llowing Lemmas. Let gcd( α, β ) b e the g reatest common d i visor of the positiv e integers α and β . Denote Q = Y p is prime p ≤ q − 1 p, for q ≥ 3 (1) and L ( s, q ) = min { l : l ≥ s and g cd( l, Q ) = 1 } , for 0 ≤ s ≤ Q − 1 } . (2) Lemma 1 : (cf. [10]) A q ( n, 3 , [ ω 0 , · · · , ω q − 1 ]) ≥  n ω 0 , · · · , ω q − 1  / ( n +Γ( t n , q )) , (3) where t n is the least n onnegative in teger such that t n ≡ n ( mo d Q ) , and Γ( t n , q ) = L ( t n , q ) − t n . ✷ Lemma 2 : (cf. [10]) Let Q be gi ven by ( 1). If gcd( n, Q ) = 1 , th en A q ( n, 3 , [ ω 0 , · · · , ω q − 1 ]) ≥  n ω 0 , · · · , ω q − 1  /n. (4) ✷ Lemma 3 : (cf. [10]) For q = 3 , A 3 ( n, 3 , [ ω 0 , ω 1 , ω 2 ]) ≥         n ω 0 , ω 1 , ω 2  /n, n = 2 k + 1 ;  n ω 0 , · · · , ω q − 1  / ( n + 1) , n = 2 k . (5) 2 ✷ In th is co rrespon dence, bound (5) is improved for ev en len gth. For a constant-co mposition code with length n , m inimum distance at least d , and constan t com position [ ω 0 , · · · , ω q − 1 ] , denote δ = ⌊ ( d − 1) / 2 ⌋ . T hen δ < ω 1 + · · · + ω q − 1 . Lemma 4 : (cf. [ 10]) For any fixed i where 0 ≤ i ≤ q − 1 , we have A q ( n, d, [ ω 0 , · · · , ω q − 1 ]) ≤  n ω 0 , · · · , ω q − 1  /  ω i + δ ω i , δ i, 0 , · · · , δ i,q − 1  (6) where δ i,j , 1 ≤ j ≤ q − 1 are nonnegative integers suc h that δ i,i = 0 , δ i, 0 + · · · + δ i,q − 1 = δ , and δ i,l ≤ ω l for 0 ≤ l ≤ q − 1 . ✷ In this paper , we show that when d = 5 , we give a lower bound have the same mag nitude with bou nd (6 ). I I I . C O N S T R U C T I O N O F C O D E S In this section , we genera lize th e con struction that is pr o- posed by X ing [16]. Let r be a prime power . W e den ote by F r the finite field with r elem ents. W e labe l all elem ents of F r F r = { α 0 = 0 , α 1 , · · · , α r − 1 } . For a positive integer m , conside r the residue ring of p olyno- mials F r [ x ] / ( x m ) . It is a finite ring and has r m elements. All invertible ele- ments of this ring form a m ultiplicative gro up, de noted b y ( F r [ x ] / ( x m )) ∗ . It is a finite a belian g roup . The quotien t group ( F r [ x ] / ( x m )) ∗ / F ∗ r is a finite ab elian g roup with r m − 1 elements. Let e is a p ositiv e integer , for a prime p , we d efine µ p ( e ) =  e if p | e ; e − 1 otherwise . Theor em 1: Let q ≥ 3 be a integer and let r be a power o f p for a prime p . If p ≥ q , then fo r any positi ve integer d 0 satisfying 1 ≤ d 0 ≤ r − 2 , there exist a q -ary ( r , M , ≥ µ p ( d 0 ) + 2 , [ ω 0 , ω 1 , · · · , ω q − 1 ]) constant- composition code with M ≥  r ω 0 , · · · , ω q − 1  /r d 0 − 1 . Proof. Consider the m ap π : V r, [ ω 0 , ω 1 , ··· , ω q − 1 ] ( q ) → ( F r [ x ] /x d 0 ) ∗ / F ∗ r ( c 1 , c 2 , · · · , c r ) 7→ r − 1 Y i =1 ( x − α i ) c i . By the Pigeonhole Principle, it is clear th at we can fin d one element f ( x ) fro m this quotient gro up such that it h as at least  r ω 0 , · · · , ω q − 1  /r d 0 − 1 pre-imag es, i.e., #( π − 1 ( f ( x ))) ≥  r ω 0 , · · · , ω q − 1  /r d 0 − 1 . Put C = π − 1 ( f ( x )) . W e are go ing to show th at C is a code with the d esired parameters. The length of C is clearly r . The remaining th ing is to sh ow that the min imum distan ce is at least µ p ( d 0 ) + 2 . Let u = ( u 1 , u 2 , · · · , u r ) and v = ( v 1 , v 2 , · · · , v r ) be two distinct co dewords of C . Th en, π ( u ) = π ( v ) = f ( x ) . This implies tha t in the gro up ( F r [ x ] / ( x d 0 )) ∗ , the element Q r − 1 i =1 ( x − α i ) u i Q r − 1 i =1 ( x − α i ) v i is eq ual to α f or som e no nzero e lement α o f F ∗ r . Put z := Q r − 1 i =1 ( x − α i ) u i Q r − 1 i =1 ( x − α i ) v i ∈ F r ( x ) . It is clear th at z is not a constant as u 6 = v .Th en the princ ipal divisor of z is e qual to div ( z ) = r − 1 X i =1 ( u i − v i ) P i + r − 1 X i =1 ( v i − u i ) ! P ∞ (7) where P i is the place corresp onding to ( x − α i ) for all 1 ≤ i ≤ r − 1 , an d P ∞ is correspo nding to th e infinite p lace. Consider the field extension F r ( x ) / F r ( z ) of degree r − 1 X i =1 | u i − v i | +      r − 1 X i =1 ( v i − u i )      ! / 2 where | . | stands for the absolu te value of a real nu mber . W e know this extension is separab le as p ≥ q (cf.[ 16]). For 1 ≤ i ≤ r − 1 , whenever u i − v i 6 = 0 , the place P i has the ramification index | u i − v i | in the extension F r ( x ) / F r ( z ) an d hence the different expon ent D P i of P i is at lea st | u i − v i | − 1 (see [1 2]). The fact that z is eq ual to α in the gro up ( F r [ x ] / ( x d 0 )) ∗ implies that P 0 is a zero of z − α with multiplicity at lea st d 0 . Hence, the ra mification ind ex of the place P 0 with respect to the extension F r ( x ) / F r ( z ) is at least d 0 , therefore , the different exp onent D P 0 ≥ d 0 − 1 . In par ticular, if p | d 0 , by Dedekind ’ s Dif ferent Theorem , we o btain D P 0 ≥ d 0 . So, D P 0 ≥ µ p ( d 0 ) . Let S = { i ∈ { 1 , 2 , · · · , r − 1 } : u i 6 = v i } and let ω be the distance betwe en u and v . 1 If u r = v r , th en | S | = ω and P r − 1 i =1 ( v i − u i ) = 0 . Hence X i ∈ S D P i ≥ X i ∈ S ( | u i − v i | − 1) = r − 1 X i =1 | u i − v i | ! − ω . By (7) th e different expon ent o f P ∞ with respect to th e extension F r ( x ) / F r ( z ) at least 0. The gen era g ( F r ( x )) 3 and g ( F r ( z )) are b oth eq ual to 0. Thus, by the Hu iwitz genus formu la (see [1 2]), we h av e − 2 = 2 g ( F r ( x )) − 2 = (2 g ( F r ( z )) − 2)[ F r ( x ) : F r ( z )] + X P D P ≥ − 2[ F r ( x ) : F r ( z )] + X i ∈ S D P i + D P 0 + D P ∞ ≥ − 2 r − 1 X i =1 | u i − v i | ! / 2 ! + r − 1 X i =1 | u i − v i | ! − ω + µ p ( d 0 ) = µ p ( d 0 ) − ω . So, ω ≥ µ p ( d 0 ) + 2 . 2 If u i 6 = v i , then | S | = ω − 1 . Hence X i ∈ S D P i ≥ X i ∈ S ( | u i − v i | − 1 ) = r − 1 X i =1 | u i − v i | ! − ω + 1 . By (7 ) the different exponen t o f P ∞ with r espect to the extension F r ( x ) / F r ( z ) at least    P r − 1 i =1 ( v i − u i )    − 1 . Thus b y th e Huiwitz g enus f ormula, we have − 2 = 2 g ( F r ( x )) − 2 = (2 g ( F r ( z )) − 2)[ F r ( x ) : F r ( z )] + X P D P ≥ − 2[ F r ( x ) : F r ( z )] + X i ∈ S D P i + D P 0 + D P ∞ ≥ − 2 r − 1 X i =1 | u i − v i | + ˛ ˛ ˛ ˛ ˛ r − 1 X i =1 ( v i − u i ) ˛ ˛ ˛ ˛ ˛ ! / 2 ! + r − 1 X i =1 | u i − v i | ! − ω + µ p ( d 0 ) + ˛ ˛ ˛ ˛ ˛ r − 1 X i =1 ( v i − u i ) ˛ ˛ ˛ ˛ ˛ − 1 = µ p ( d 0 ) − ω . So, ω ≥ µ p ( d 0 ) + 2 . The d esired result follows. ✷ I V . S O M E E X A M P L E S F O R L OW E R B O U N D O N C O N S T A N T - C O M P O S I T I O N C O D E S Now , we can get some im proved lo wer boun ds for co nstant- composition cod es fro m Theore m 1. W e adopt the no tations and termin ologies in the pr evious section a nd consider the quotient g roup ( F r [ x ] /x d 0 ) ∗ / F ∗ r . Example 1 . Consider d 0 = 2 (1) For the ca se p ≥ q ≥ 3 , µ p ( d 0 ) = 1 , the gro up ( F r [ x ] / ( x 2 )) ∗ / ( F r ) ∗ has r elements. By Th eorem 1 we can get a constan t-compo sition code with par ameters ( r , M , d, [ ω 0 , ω 1 , · · · , ω q − 1 ]) , where d ≥ 3 , an d M ≥  r ω 0 , · · · , ω q − 1  /r. Then we ob tain A q ( r , 3 , [ ω 0 , · · · , ω q − 1 ]) ≥  r ω 0 , · · · , ω q − 1  /r. (8) The bound in this case ach ieves the one gi ven in Lemma 2 fo r co des with o dd leng th. (2) Now we c onsider the code of ev en leng th. Let q = 3 , 2 | r , from the first part proof of the orem 1, we kn ow that we can get a constant-co mposition code o f size ≥  r ω 0 , ω 1 , ω 2  /r , then we want to sh ow this co de has minimum distance ≥ 3 . For two distinct codew ords u = ( u 1 , u 2 , · · · , u r ) and v = ( v 1 , v 2 , · · · , v r ) , similar to Theorem 1, co nsider u ( x ) v ( x ) := Q r − 1 i =1 ( x − α i ) u i Q r − 1 i =1 ( x − α i ) v i ≡ α mod( x 2 ) . (9) for some n onzero elemen t α of F ∗ r . 1 If u r 6 = v r , the distance between u and v is 2 if an d only if u ( x ) v ( x ) = ( x − α i ) o r u ( x ) v ( x ) = 1 x − α i for so me i, 1 ≤ i ≤ r − 1 . Both of th ese two cases are not satisfy (9), so we ge t d ≥ 3 . 2 If u r = v r , it is easy to know that the distance be- tween u = ( u 1 , u 2 , · · · , u r ) and v = ( v 1 , v 2 , · · · , v r ) is 2 if and o nly if u ( x ) v ( x ) = x − α i x − α j or u ( x ) v ( x ) = ( x − α i ) 2 ( x − α j ) 2 , for some i, j, 1 ≤ i 6 = j ≤ r − 1 . Since char F r = 2 , both of th ese two cases ar e n ot satisfy (9) , so we get d ≥ 3 . Then A 3 ( r , 3 , [ ω 0 , ω 1 , ω 2 ]) ≥  r ω 0 , ω 1 , ω 2  /r. (10) Bound (10) impr oves the one given in Lem ma 3 when the len gth o f co de is even. Example 2 . Consider d 0 = 3 : (1) For th e case p = char ( F r ) = q = 3 . Th en p | d 0 , since µ p ( d 0 ) = 3 we get d ≥ 5 . By Theorem 1, we g et a 3 - ary ( r , M , 5 , [ ω 0 , ω 1 , ω 2 ]) co nstant-com position code, where M ≥  r ω 0 , ω 1 , ω 2  /r 2 . Hence, A 3 ( r , 5 , [ ω 0 , ω 1 , ω 2 ]) ≥  r ω 0 , ω 1 , ω 2  /r 2 . (11) Lemma 3 g i ven a uppe r bound of con stant-comp osition codes. No w we tak e d = 5 , then δ = ⌊ d − 1 2 ⌋ = 2 , it is easy to know th at th ere exist ω i ≥ ⌊ r /q ⌋ for 0 ≤ i ≤ q − 1 . So we have A 3 ( r , 5 , [ ω 0 , ω 1 , ω 2 ]) ≤  n ω 0 , ω 1 , ω 2  /  ω i + 2 ω i , δ i, 0 , δ i, 1 , δ i, 2  where δ i,j are nonn egati ve δ i, 0 + δ i, 1 + δ i, 2 = 2 , we choose δ i, 0 , δ i, 1 , δ i, 2 such that  2 δ i, 0 , δ i, 1 , δ i, 2  = 2 , then t ( r ) =  ω i + 2 ω i , δ i, 0 , δ i, 1 , δ i, 2  = ( ω i + 2)( ω i + 1) ≥ ( r q + 1) r q = O ( r 2 ) wh en r → ∞ , then we obtain an upper boun d for constant composition code over F 3 of 4 minimum distance 5 A 3 ( r , 5 , [ ω 0 , ω 1 , ω 2 ]) ≤  r ω 0 , ω 1 , ω 2  /t ( r ) where t ( r ) = O ( r 2 ) , co mpare this upper b ound with our lo wer bound in (9), our lower bound given the best possible size u p to m agnitude. (2) For the case p ≥ q ≥ 3 and p > 3 , then we o btain d ≥ 4 since µ p ( d 0 ) = 2 , By Th eorem 1, we obtain a q -ar y ( r , M , 4 , [ ω 0 , ω 1 , · · · , ω q − 1 ]) constan t-compo sition code, wh ere M ≥  r ω 0 , · · · , ω q − 1  /r 2 , and the lower bo und A q ( r , 4 , [ ω 0 , · · · , ω q − 1 ]) ≥  r ω 0 , · · · , ω q − 1  /r 2 . (12) Example 3 . Let d 0 = 5 (1) For the case p = q = 5 , we get d ≥ 7 , then by Theorem 1, A q ( r , 7 , [ ω 0 , · · · , ω q − 1 ]) ≥  r ω 0 , · · · , ω q − 1  /r 4 . (2) If p = > 5 an d 3 ≤ q ≤ p , we get d ≥ 6 an d lower bound A q ( r , 6 , [ ω 0 , · · · , ω q − 1 ]) ≥  r ω 0 , · · · , ω q − 1  /r 4 . Remark 2: 1 Th e constru ction in this p aper pr oduces a lower bound o n constant-co mposition co des for arbitrary gi ven m inimum distance. 2 As far as we know , excep t f or the bo und giv en in this paper, th ere are no b ound s on A q ( n, d, [ ω 0 , · · · , ω q − 1 ]) , where d ≥ 4 , so far . 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