Uniform constant composition codes derived from repeated‐root cyclic codes

Abstract

Uniform constant composition (UCC) codes are introduced, which are p-ary constant composition codes with block length mp where each p-ary symbol appears exactly m times in each codeword, being m a positive integer and p a prime. UCC codes are derived from a new class of p-ary repeated-root cyclic codes and two constructions are presented. Both constructions are at least asymptotically optimal with increasing p when compared with known best bounds for constant composition codes. Introduction: A constant composition code [1] is a block code defined over an alphabet with q symbols, denoted as 1, 2,. .. , q, the codebook of which consists of codewords where the symbol i occurs w i times, 1 ≤ i ≤ q, and is usually a non-linear code. We define a uniform constant composition (UCC) code as a constant composition code of block length mq, where m is a positive integer, for which w i = m, 1 ≤ i ≤ q. In this Letter, we consider only p-ary UCC codes of block length mp, where p is a prime, m is a positive integer, m ò 0 mod p, i.e. m is not a multiple of p, since the codes resulting when m ; 0 mod p represent a straightforward generalisation. Constant composition codes have recently been considered for important applications requiring non-binary alphabets [2] as, for example, power-line communications [3], construction of DNA codes [4], multiple-access communications and frequency hopping [5]. In order to construct UCC codes, we first introduce a class of p-ary (n, k, d) repeated-root cyclic codes with block length n = mp, k information symbols and minimum distance d. Two code constructions are presented, each one characterised by the corresponding p-ary generator polynomial g(x), which is a factor of x n − 1, with the remark that n = mp is the smallest value of n for which g(x) divides x n − 1. Construction 1 produces (mp, 2, m(p − 1)) codes which have a subset of M 1 = p(p − 1) UCC codewords, while Construction 2 produces (mp, m + 1, d) codes, with d = p − 1 for m = 1 and d = p for m ≥ 2, which have a subset of M 2 = p m (p − 1) UCC codewords.

On the cyclic order distribution and partitioning of linear cyclic codes

“…As ongoing research, the authors are investigating linear cyclic codes in R q,n , where q and n are not necessarily coprime, i.e., gcd(q, n) ≠ 1 , which leads to the class of repeated-root cyclic codes [3]. This class of codes has been subject of research for years, and recently it was used in order to construct uniform composition codes [6].…”

“…(x) = x 11 + x 10 + x 9 + x 8 + x6 + x 4 + x 3 + 1, c 2 (x) = x 14 + x 13 + x 12 + x 7 + x 5 + x 2 + x, c 3 (x) = x 13 + x 12 + x 11 + x 8 + x 7 + x, c 4 (x) = x 14 + x 13 + x 9 + x 7 + x 6 + x 5 + x 3 + x 2 + x + 1,c 5 (x) = x 14 + x 12 + x 10 + x 5 + x 4 + x 2 , c 6 (x) = x 14 + x 10 + x 9 + x 6 + x 5 + x 4 + x 3 + x 2 + 1, c 7 (x) = x 12 + x 11 + x 10 + x 8 + x 4 and c 8 (x) = x 13 + x 11 + x 9 + x 8 + x 7 + x 6 + x 3 + x + 1.…”

MOCZ for Blind Short-Packet Communication: Basic Principles

“…Note, the cardinality of a cycling register code C CR is minimal if n is prime. This can be seen by acknowledging the fact that the cyclic order of a codeword is always a divisor of n, see for example [24,Lem.1]. Therefore, if n is prime we only have the trivial orders 1 and n, where the only codewords with order ν = 1 are the all one 1 and all zero 0 codeword and all other codewords have the maximal or full cyclic order ν = n. Hence, extracting the CRC from F n 2 obtains the same cardinality (41).…”

Construction of optimal uniform constant composition codes

“…When , this code is called permutation code . As stated in [1], these UCC codes may be applied in DNA code constructions and multiple‐access communications, just citing some applications.…”

“…Recently, uniform constant composition codes (UCC‐codes) have been introduced by da Rocha et al . [1] and are particular cases of constant composition codes [2]. A ‐UCC‐code is composed by N ‐length codewords, whose minimum (Hamming) distance among them is equal to D , and each one of the q possible symbols appears times in each codeword.…”