Bounds and Constructions for Optimal Constant Weight Conflict-Avoiding Codes

2007 IEEE International Symposium on Information Theory Pub Date : 2007-06-24 DOI:10.1109/ISIT.2007.4557248

K. Momihara, Meinard Müller, Junya Satoh, Masakazu Jimbo

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Abstract

A conflict-avoiding code (CAC) C of length n with weight k is a family of binary sequences of length n and weight k satisfying Sigma₀ _les _t _les _n-1 x_it x_j, _t+s les lambda for any distinct codewords x_j = (x_i0,x_i1,hellip,x_i, _n-1) and x_j = (x_j0, x_j1,hellip, x_j, _n-1) in C and for any integer s, where the subscripts are taken modulo n. A CAC with maximal code size for given n and k is said to be optimal. A CAC has been studied for sending messages correctly through a multiple-access channel. The use of an optimal CAC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, the case lambda = 1 is treated, and various direct and recursive constructions of optimal CACs for weight k = 4 and 5 are obtained by providing constructions of CACs for general weight k. In particular, the maximum code size of CACs satisfying certain sufficient conditions is determined through number theoretical and combinatorial approaches.

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最优等权避免冲突码的界和构造

conflict-avoiding代码长度(CAC) C n重量k是一个家庭的二进制序列的长度n和k重量满足Sigma0 les t n - 1——xit xj, t + s lesλ为任何不同的码字xj = (xi0ξ1,白马王子,xi, n - 1)和xj = (xj0 xj1,白马王子,xj, n - 1)在C和任何整数年代,下标的模n。一个CAC与最大代码大小对于给定n和k据说是最优的。研究了通过多址信道正确发送消息的CAC。使用最佳CAC可以使尽可能多的异步用户高效可靠地传输信息。本文处理了λ = 1的情况，通过给出一般权值k的最优ccs的构造，得到了权值k = 4和5的最优ccs的各种直接和递归构造，特别是通过数论和组合的方法确定了满足一定充分条件的ccs的最大码长。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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2007 IEEE International Symposium on Information Theory

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