aa r X i v : . [ c s . I T ] F e b A Construction for Constant-Composition Codes
Yang Ding
Abstract — By employing the residue polynomials, we give aconstruction of constant-composition codes. This constructiongeneralizes the one proposed by Xing [16]. It turns out thatwhen d = 3 this construction gives a lower bound of constant-composition codes improving the one in [10] for some case.Moreover, for d > , we give a lower bound on maximal size ofconstant-composition codes. In particular, our bound for d = 5 gives the best possible size of constant-composition codes up tomagnitude. Index Terms — constant-composition codes, genus, residue poly-nomial, rational function fields.
I. INTRODUCTIONConstant-composition codes are a subclass of constantweight codes, in which both weight restrict and elementcomposition restrict are involved. The class of constant-composition codes have attracted recent interest due to itsnumerous applications, such as in determining the zero errordecision feedback capacity of discrete memoryless channels[15], multiple-access communications [8], spherical codes formodulation [9], DNA codes [11], powerline communications[2], and frequency hopping [3].One of the most fundamental problem in coding theory isthe problem of determining the maximum size of a blockcode, given its length and minimum distance. The problem ofdetermining the maximum size of a constant-composition codeis much less understood than the constant-weight and linearcases. In the recent years, researches consider the problems ofmaximizing the size of a constant-composition code (see [1],[10], [13]), and constructing optimal codes to achieve thesebounds (see [4], [5], [6], [7], [14]). In this paper, we givea construction for constant-composition codes then produce alower bound on constant-composition codes for arbitrary givenminimum distance. We show that when q = 3 and d = 5 , ourbound gives the best possible size of constant-compositioncodes up to magnitude. As far as we know, except for thebound given in this paper, there is no bounds on d > so far.This correspondence is organized as follows. In SectionII, we introduce some basic definitions and notations. Wealso review some basic properties which will be used in thiscorrespondence. The main construction is presented in SectionIII. In Section IV, Theorem 1 in section II are used to obtainsome good lower bounds on constant-composition codes. This work was supported by the China Scholarship Council.The author is with the Department of Mathematics, SoutheastUniversity, Nanjing, 210096, People’s Republic of China (e-mail:[email protected]). This work was carried out while the authorwas studying in Division of Mathematical Sciences, School of Physical andMathematical Sciences, Nanyang Technological University, Singapore underthe exchange program.
II. PRELIMINARYWe use the standard notations for codes as follows. LetZ q denote the set { , , · · · , q − } , and let Z nq be theset of all n -tuples over Z q , where q is a positive inte-ger. Let V n, [ ω , ω , ··· , ω q − ] ( q ) denote the set of n -tuplesover Z q of the fixed composition [ ω , ω , · · · , ω q − ] ,i.e., the number of ’s, ’s, · · · , q − ’s in the n -tuple over Z q is given by ω , ω , · · · , ω q − , respectively,where n = ω + ω + · · · + ω q − . It is obvious that V n, [ ω , ω , ··· , ω q − ] ( q ) contains (cid:18) nω , ω , · · · , ω q − (cid:19) ele-ments. An ( n, M, d, [ ω , ω , · · · , ω q − ]) q constant-composition code C is a subset of V n, [ ω , ω , ··· , ω q − ] ( q ) with size M and minimum Hamming distance d . We use A q ( n, d, [ ω , ω , · · · , ω q − ]) to denote the maximum sizeof an ( n, M, d, [ ω , ω , · · · , ω q − ]) q constant-compositioncode.In order to establish our results in this correspondence, weneed the following Lemmas.Let gcd( α, β ) be the greatest common divisor of the positiveintegers α and β . Denote Q = Y p is prime p ≤ q − p, for q ≥ (1)and L ( s, q ) = min { l : l ≥ s and gcd( l, Q ) = 1 } , for 0 ≤ s ≤ Q − } . (2) Lemma 1: (cf. [10]) A q ( n, , [ ω , · · · , ω q − ]) ≥ (cid:18) nω , · · · , ω q − (cid:19) / ( n +Γ( t n , q )) , (3)where t n is the least nonnegative integer such that t n ≡ n (mod Q ) , and Γ( t n , q ) = L ( t n , q ) − t n . ✷ Lemma 2: (cf. [10]) Let Q be given by (1). If gcd( n, Q ) =1 , then A q ( n, , [ ω , · · · , ω q − ]) ≥ (cid:18) nω , · · · , ω q − (cid:19) /n. (4) ✷ Lemma 3: (cf. [10]) For q = 3 , A ( n, , [ ω , ω , ω ]) ≥ (cid:18) nω , ω , ω (cid:19) /n, n = 2 k + 1 ; (cid:18) nω , · · · , ω q − (cid:19) / ( n + 1) , n = 2 k . (5) ✷ In this correspondence, bound (5) is improved for even length.For a constant-composition code with length n , minimumdistance at least d , and constant composition [ ω , · · · , ω q − ] ,denote δ = ⌊ ( d − / ⌋ . Then δ < ω + · · · + ω q − . Lemma 4: (cf. [10]) For any fixed i where ≤ i ≤ q − ,we have A q ( n, d, [ ω , · · · , ω q − ]) ≤ (cid:18) nω , · · · , ω q − (cid:19) / (cid:18) ω i + δω i , δ i, , · · · , δ i,q − (cid:19) (6)where δ i,j , ≤ j ≤ q − are nonnegative integers suchthat δ i,i = 0 , δ i, + · · · + δ i,q − = δ , and δ i,l ≤ ω l for ≤ l ≤ q − . ✷ In this paper, we show that when d = 5 , we give a lowerbound have the same magnitude with bound (6).III. CONSTRUCTION OF CODESIn this section, we generalize the construction that is pro-posed by Xing [16]. Let r be a prime power. We denote by F r the finite field with r elements. We label all elements of F r F r = { α = 0 , α , · · · , α r − } . For a positive integer m , consider the residue ring of polyno-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 multiplicative group, denoted by ( F r [ x ] / ( x m )) ∗ . It is a finite abelian group. The quotient group ( F r [ x ] / ( x m )) ∗ / F ∗ r is a finite abelian group with r m − elements.Let e is a positive integer, for a prime p , we define µ p ( e ) = (cid:26) e if p | e ; e − otherwise . Theorem 1:
Let q ≥ be a integer and let r be apower of p for a prime p . If p ≥ q , then for any positiveinteger d satisfying ≤ d ≤ r − , there exist a q -ary ( r, M, ≥ µ p ( d ) + 2 , [ ω , ω , · · · , ω q − ]) constant-composition code with M ≥ (cid:18) rω , · · · , ω q − (cid:19) /r d − . Proof.
Consider the map π : V r, [ ω , ω , ··· , ω q − ] ( q ) → ( F r [ x ] /x d ) ∗ / F ∗ r ( c , c , · · · , c r ) r − Y i =1 ( x − α i ) c i . By the Pigeonhole Principle, it is clear that we can find oneelement f ( x ) from this quotient group such that it has at least (cid:18) rω , · · · , ω q − (cid:19) /r d − pre-images, i.e., π − ( f ( x ))) ≥ (cid:18) rω , · · · , ω q − (cid:19) /r d − . Put C = π − ( f ( x )) . We are going to show that C is a code with the desiredparameters. The length of C is clearly r . The remaining thingis to show that the minimum distance is at least µ p ( d ) + 2 .Let u = ( u , u , · · · , u r ) and v = ( v , v , · · · , v r ) be twodistinct codewords of C . Then, π ( u ) = π ( v ) = f ( x ) . Thisimplies that in the group ( F r [ x ] / ( x d )) ∗ , the element Q r − i =1 ( x − α i ) u i Q r − i =1 ( x − α i ) v i is equal to α for some nonzero element α of F ∗ r .Put z := Q r − i =1 ( x − α i ) u i Q r − i =1 ( x − α i ) v i ∈ F r ( x ) . It is clear that z is not a constant as u = v .Then the principaldivisor of z is equal todiv ( z ) = r − X i =1 ( u i − v i ) P i + r − X i =1 ( v i − u i ) ! P ∞ (7)where P i is the place corresponding to ( x − α i ) for all ≤ i ≤ r − , and P ∞ is corresponding to the infinite place.Consider the field extension F r ( x ) / F r ( z ) of degree r − X i =1 | u i − v i | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − X i =1 ( v i − u i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! / where | . | stands for the absolute value of a real number. Weknow this extension is separable as p ≥ q (cf.[16]).For ≤ i ≤ r − , whenever u i − v i = 0 , the place P i has theramification index | u i − v i | in the extension F r ( x ) / F r ( z ) andhence the different exponent D P i of P i is at least | u i − v i | − (see [12]).The fact that z is equal to α in the group ( F r [ x ] / ( x d )) ∗ implies that P is a zero of z − α with multiplicity at least d .Hence, the ramification index of the place P with respectto the extension F r ( x ) / F r ( z ) is at least d , therefore, thedifferent exponent D P ≥ d − . In particular, if p | d , byDedekind’s Different Theorem, we obtain D P ≥ d . So, D P ≥ µ p ( d ) .Let S = { i ∈ { , , · · · , r − } : u i = v i } and let ω be the distance between u and v .1 If u r = v r , then | S | = ω and P r − i =1 ( v i − u i ) = 0 . Hence X i ∈ S D P i ≥ X i ∈ S ( | u i − v i | −
1) = r − X i =1 | u i − v i | ! − ω. By (7) the different exponent of P ∞ with respect to theextension F r ( x ) / F r ( z ) at least 0. The genera g ( F r ( x )) and g ( F r ( z )) are both equal to 0. Thus, by the Huiwitzgenus formula (see [12]), we have − g ( F r ( x )) −
2= (2 g ( F r ( z )) − F r ( x ) : F r ( z )] + X P D P ≥ − F r ( x ) : F r ( z )] + X i ∈ S D P i + D P + D P ∞ ≥ − r − X i =1 | u i − v i | ! / ! + r − X i =1 | u i − v i | ! − ω + µ p ( d )= µ p ( d ) − ω. So, ω ≥ µ p ( d ) + 2 .2 If u i = v i , then | S | = ω − . Hence X i ∈ S D P i ≥ X i ∈ S ( | u i − v i | −
1) = r − X i =1 | u i − v i | ! − ω + 1 . By (7) the different exponent of P ∞ with respect tothe extension F r ( x ) / F r ( z ) at least (cid:12)(cid:12)(cid:12)P r − i =1 ( v i − u i ) (cid:12)(cid:12)(cid:12) − .Thus by the Huiwitz genus formula, we have − g ( F r ( x )) −
2= (2 g ( F r ( z )) − F r ( x ) : F r ( z )] + X P D P ≥ − F r ( x ) : F r ( z )] + X i ∈ S D P i + D P + D P ∞ ≥ − r − X i =1 | u i − v i | + ˛˛˛˛˛ r − X i =1 ( v i − u i ) ˛˛˛˛˛! / ! + r − X i =1 | u i − v i | ! − ω + µ p ( d ) + ˛˛˛˛˛ r − X i =1 ( v i − u i ) ˛˛˛˛˛ − µ p ( d ) − ω. So, ω ≥ µ p ( d ) + 2 .The desired result follows. ✷ IV. SOME EXAMPLES FOR LOWER BOUND ONCONSTANT-COMPOSITION CODESNow, we can get some improved lower bounds for constant-composition codes from Theorem 1. We adopt the notationsand terminologies in the previous section and consider thequotient group ( F r [ x ] /x d ) ∗ / F ∗ r . Example 1 . Consider d = 2 (1) For the case p ≥ q ≥ , µ p ( d ) = 1 , the group ( F r [ x ] / ( x )) ∗ / ( F r ) ∗ has r elements. By Theorem 1 wecan get a constant-composition code with parameters ( r, M, d, [ ω , ω , · · · , ω q − ]) , where d ≥ , and M ≥ (cid:18) rω , · · · , ω q − (cid:19) /r. Then we obtain A q ( r, , [ ω , · · · , ω q − ]) ≥ (cid:18) rω , · · · , ω q − (cid:19) /r. (8) The bound in this case achieves the one given in Lemma2 for codes with odd length.(2) Now we consider the code of even length. Let q = 3 , | r , from the first part proof of theorem 1, we knowthat we can get a constant-composition code of size ≥ (cid:18) rω , ω , ω (cid:19) /r , then we want to show this codehas minimum distance ≥ . For two distinct codewords u = ( u , u , · · · , u r ) and v = ( v , v , · · · , v r ) , similar toTheorem 1, consider u ( x ) v ( x ) := Q r − i =1 ( x − α i ) u i Q r − i =1 ( x − α i ) v i ≡ α mod( x ) . (9)for some nonzero element α of F ∗ r .1 If u r = v r , the distance between u and v is 2 if andonly if u ( x ) v ( x ) = ( x − α i ) or u ( x ) v ( x ) = x − α i for some i, ≤ i ≤ r − . Both of these two cases are notsatisfy (9), so we get d ≥ .2 If u r = v r , it is easy to know that the distance be-tween u = ( u , u , · · · , u r ) and v = ( v , v , · · · , v r ) is if and only if u ( x ) v ( x ) = x − α i x − α j or u ( x ) v ( x ) = ( x − α i ) ( x − α j ) ,for some i, j, ≤ i = j ≤ r − . Since char F r = 2 ,both of these two cases are not satisfy (9) , so we get d ≥ .Then A ( r, , [ ω , ω , ω ]) ≥ (cid:18) rω , ω , ω (cid:19) /r. (10)Bound (10) improves the one given in Lemma whenthe length of code is even. Example 2 . Consider d = 3 :(1) For the case p = char ( F r ) = q = 3 . Then p | d , since µ p ( d ) = 3 we get d ≥ . By Theorem 1, we get a -ary ( r, M, , [ ω , ω , ω ]) constant-composition code, where M ≥ (cid:18) rω , ω , ω (cid:19) /r . Hence, A ( r, , [ ω , ω , ω ]) ≥ (cid:18) rω , ω , ω (cid:19) /r . (11)Lemma 3 given a upper bound of constant-compositioncodes. Now we take d = 5 , then δ = ⌊ d − ⌋ = 2 , it is easyto know that there exist ω i ≥ ⌊ r/q ⌋ for ≤ i ≤ q − .So we have A ( r, , [ ω , ω , ω ]) ≤ (cid:18) nω , ω , ω (cid:19) / (cid:18) ω i + 2 ω i , δ i, , δ i, , δ i, (cid:19) where δ i,j are nonnegative δ i, + δ i, + δ i, = 2 , wechoose δ i, , δ i, , δ i, such that (cid:18) δ i, , δ i, , δ i, (cid:19) = 2 ,then t ( r ) = (cid:18) ω i + 2 ω i , δ i, , δ i, , δ i, (cid:19) = ( ω i + 2)( ω i + 1) ≥ ( rq + 1) rq = O ( r ) when r → ∞ , then we obtain anupper bound for constant composition code over F of minimum distance A ( r, , [ ω , ω , ω ]) ≤ (cid:18) rω , ω , ω (cid:19) /t ( r ) where t ( r ) = O ( r ) , compare this upper bound withour lower bound in (9), our lower bound given the bestpossible size up to magnitude.(2) For the case p ≥ q ≥ and p > , then we obtain d ≥ since µ p ( d ) = 2 , By Theorem 1, we obtain a q -ary ( r, M, , [ ω , ω , · · · , ω q − ]) constant-compositioncode, where M ≥ (cid:18) rω , · · · , ω q − (cid:19) /r , and the lower bound A q ( r, , [ ω , · · · , ω q − ]) ≥ (cid:18) rω , · · · , ω q − (cid:19) /r . (12) Example 3 . Let d = 5 (1) For the case p = q = 5 , we get d ≥ , then by Theorem 1, A q ( r, , [ ω , · · · , ω q − ]) ≥ (cid:18) rω , · · · , ω q − (cid:19) /r . (2) If p = > and ≤ q ≤ p , we get d ≥ and lower bound A q ( r, , [ ω , · · · , ω q − ]) ≥ (cid:18) rω , · · · , ω q − (cid:19) /r . Remark 2: A q ( n, d, [ ω , · · · , ω q − ]) ,where d ≥ , so far.A CKNOWLEDGMENT
The author is grateful to Profs. Keqin Feng, Jianlong Chenand Chaoping Xing for their guidance.R
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