Cyclically permutable codes specified by roots of generator polynomial

Abstract

A condition is proven on the roots of the generator polynomial of a q-ary cyclic code of block length n which guarantees full cyclic order for all nonzero codewords. For such cyclic codes having n = q m − 1, a theorem is proven which allows to partition the set of all nonzero codewords into cyclic equivalence classes of order n.Introduction: In this Letter, a condition is proven on the roots of the generator polynomial of a linear q-ary cyclic code C of block length n which guarantees full cyclic order for all nonzero codewords, where q is the power of a prime and n is a positive integer which divides q m − 1. For such cyclic codes having n = q m − 1, a theorem is proven which allows a systematic partition of the set of all nonzero codewords of C into cyclic equivalence classes of order n [1]. We refer to this latter result as the separation theorem. The goal of the theory in this Letter is the construction of 'cyclically permutable codes' [2].Binary cyclically permutable codes have many applications as, for example, protocol sequences for the collision channel without feedback [1,3], as well as improving robustness against clipping attacks of watermarking systems [4]. Additionally, non-binary cyclically permutable codes have applications in direct sequence code division multiple access (DS-CDMA) systems with asynchronous base stations [5].In [1], conditions are given that should be obeyed by a construction of a cyclically permutable code. The first condition is that the technique for specifying equivalence classes must be easy to build. The second condition specifies the efficiency of the construction, namely, for a given a q-ary cyclic code with block length n, having q k codewords, the number of distinct cyclic equivalence classes produced should be close to the maximum value of q k /n. A third desirable condition is that the minimum Hamming distance of the cyclically permutable code should approach the maximum possible value for the given code parameters. In the sequel, we show that the proposed approach for generating cyclically permutable codes is easily implementable, and for linear q-ary cyclic codes it is optimal in the sense that the number of cyclic equivalence classes selected is precisely (q k − 1)/n, obtained by removing the allzero codewords because it has cyclic order 1 (see Definition 1).

MOCZ for Blind Short-Packet Communication: Basic Principles

“…However, the construction from F n 2 , even for n prime, is a combinatorial problem, especially the decoding. Hence, to reduce the combinatorial complexity, many approaches starting from cyclic codes and extract all codewords with maximal cyclic order [28]- [30]. Since a cyclic (n, k, d min ) code corrects up to (d min − 1)/2 bit errors, any CPC code extraction will inherit the error correction capability.…”

“…However, the authors want to emphasize that other CPC constructions exist which do not require Mersenne prime lengths n or even binary alphabets. In [30] the authors showed construction of CPCs with alphabet size q, given as a power of a prime, and block length n = Let us mention the shortest non-trivial CPC, which can be addressed without cyclic code construction, but also with no error correction capabilities.…”

Códigos constacı́clicos, códigos ciclicamente permutáveis e sequências de protocolo para o canal de colisão sem realimentação

“…Conforme justificado na sequência,é possível particionar um código constacíclico em classes de equivalência constacíclica de modo semelhante ao que foi feito com códigos cíclicos em [15]. Neste caso, considere duas palavras-código c 1 (x) e c 2 (x) pertencentes a um código constacíclico pario C cujas palavras-código são reduzidas mod(x p+1 − a).…”

“…Este artigo propõe um novo conjunto de códigos CP, derivados de códigos constacíclicos p-ários, cujas palavras-código podem ser usadas como sequências de protocolo para o CCsR. Os teoremas 1 e 2 são a versão para códigos constacíclicos dos teoremas 1 e 2 em [15] para códigos cíclicos. A Construção 1éó tima no sentido que atinge precisamente o limitante superior para o número de classes de equivalência constacíclica dos códigos usados.…”

On the cyclic order distribution and partitioning of linear cyclic codes

“…However, the linear cyclic codes considered are restricted to be binary linear cyclic codes of block length n = 2 m − 1 , a Mersenne prime. In [11] qary linear cyclic codes with block length n = q m − 1 , where q is a prime power, were effectively used to generate CPC's. The results in [11] increase the range of linear cyclic codes that can be used to generate CPC's.…”