Explicit Cyclic and Quasi-Cyclic Codes With Optimal, Best Known Parameters, and Large Relative Minimum Distances

IF 2.2 3区 计算机科学 Q3 COMPUTER SCIENCE, INFORMATION SYSTEMS
Conghui Xie;Hao Chen;Chen Yuan
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引用次数: 0

Abstract

In this paper, we construct many infinite families of distance-optimal codes with new parameters, some of which are BCH codes and quasi-cyclic codes. In particular, we report the first infinite family of binary distance-optimal BCH codes with the minimum distance 8. Secondly, several infinite families of binary BCH codes and quasi-cyclic codes are presented. Many codes in these families have optimal or best known parameters. Thirdly, we construct infinite families of binary cyclic $\left [{{n, \geq \frac {n+1}{2},d}}\right]_{2}$ codes with minimum distances $d \geq \lceil \frac {n-1}{\prod _{i=1}^{s}p_{i}}\rceil $ , $n=(2^{p_{1}}-1)(2^{p_{2}}-1) \cdots (2^{p_{s}}-1)$ , $p_{1}, \ldots, p_{s}$ are different primes. Our construction extends the main result of a recent paper published by Sun et al. to much more general binary cyclic codes with various lengths. We also construct an infinite family of binary quasi-cyclic codes with the rate around $\frac {1}{2}$ and relative minimum distance lower bounded by $O\left ({{\frac {1}{\log _{2} \log _{2} n}}}\right)$ .
具有最优、已知参数和较大相对最小距离的显式循环和拟循环码
本文构造了许多具有新参数的无限族距离最优码,其中一些是BCH码和拟循环码。特别地,我们报道了第一个无限族的二进制距离最优BCH码,最小距离为8。其次,给出了几个无限族的二进制BCH码和拟循环码。这些家族中的许多代码具有最优或最知名的参数。第三,我们构造了无限族的二进制循环$\left [{{n, \geq \frac {n+1}{2},d}}\right]_{2}$码,其最小距离$d \geq \lceil \frac {n-1}{\prod _{i=1}^{s}p_{i}}\rceil $, $n=(2^{p_{1}}-1)(2^{p_{2}}-1) \cdots (2^{p_{s}}-1)$, $p_{1}, \ldots, p_{s}$是不同的素数。我们的构造将Sun等人最近发表的一篇论文的主要结果扩展到具有各种长度的更通用的二进制循环码。我们还构造了一个无限族的二进制拟循环码,其速率约为$\frac {1}{2}$,相对最小距离下界为$O\left ({{\frac {1}{\log _{2} \log _{2} n}}}\right)$。
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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来源期刊
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory 工程技术-工程:电子与电气
CiteScore
5.70
自引率
20.00%
发文量
514
审稿时长
12 months
期刊介绍: The IEEE Transactions on Information Theory is a journal that publishes theoretical and experimental papers concerned with the transmission, processing, and utilization of information. The boundaries of acceptable subject matter are intentionally not sharply delimited. Rather, it is hoped that as the focus of research activity changes, a flexible policy will permit this Transactions to follow suit. Current appropriate topics are best reflected by recent Tables of Contents; they are summarized in the titles of editorial areas that appear on the inside front cover.
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