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数学进展 - 2020, Vol. 49(4): 481-496
研究论文
环${\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+\cdots+u^{k-1}\mathbb{F}_{p^m}}$ ${(u^k=0)}$上基于定义集的几类线性码
Several Classes of Linear Codes from Defining Sets over ${\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+\cdots+u^{k-1}\mathbb{F}_{p^m}\ (u^k=0)}$

朱灿泽1, 方颖珏1, 廖群英2
ZHU Canze1, FANG Yingjue1, LIAO Qunying2,*

1.深圳大学数学与统计学院, 深圳, 广东, 518060;
2.四川师范大学数学科学学院, 成都, 四川, 610066
1. School of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong, 518060, P. R. China;
2. School of Mathematical Sciences, Sichuan Normal University, Chengdu, Sichuan, 610066, P. R. China

收稿日期: 2019-05-31
出版日期: 2020-08-11
2020, Vol. 49(4): 481-496
DOI: 10.11845/sxjz.2019058b


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摘要 本文研究了有限链环$\mathcal{R}_k=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+\cdots +u^{k-1}\mathbb{F}_{p^m}(u^k=0)$ 上基于定义集的线性码的李权重分布, 其中$p$是奇素数,$m,k$是正整数. 通过使用特征和工具,我们计算出上述线性码在Gray映射下的完备权重计数器. 此外,还提出了一类满足Griesmer界的常数权重码,它可用于构造满足LVFC界限的最优常维码, 本文推广了文献[IEEE Trans. Inf. Theory, 2015, 61(11): 5835-5842]和[Des. Codes Cryptogr., 2019, 87(1): 15-29]的主要结果.
关键词 线性码有限链环完全权重计数器特征和    
Abstract:In this paper, we study the Lee-weight distribution of linear codes from defining sets over the finite chain ring $\mathcal{R}_k=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+\cdots+u^{k-1}\mathbb{F}_{p^m}\ (u^k=0)$, where $p$ is an odd prime, $m$, $k$ are positive integers. We establish the complete weight enumerator for the images of these linear codes under the Gray map explicitly by using character sums over finite fields. Furthermore, a class of constant weight codes achieving the Griesmer bound which can be used to construct optimal constant composition codes meeting the LVFC bound is obtained, which generalizes the main results in [IEEE Trans. Inf. Theory, 2015, 61(11): 5835-5842] and [Des. Codes Cryptogr., 2019, 87(1): 15-29].
Key wordslinear code    finite chain ring    complete weight enumerator    character sums
PACS:  O157.4  
通讯作者: E-mail: * qunyingliao@sicnu.edu.cn   
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