有限链环上循环码的重量分布  

Weight Distributions of Cyclic Codes over Finite Chain Rings

作　　者：郑雨琦 孟祥蕊 高健 ZHENG Yuqi;MENG Xiangrui;GAO Jian(School of Mathematics and Statistics,Shandong University of Technology,Zibo 255000;Chern Institute of Mathematics and LPMC,Nankai University,Tianjin 300071)

机构地区：[1]山东理工大学数学与统计学院,淄博255000 [2]南开大学陈省身数学研究所,天津300071

出　　处：《系统科学与数学》2024年第9期2850-2864,共15页Journal of Systems Science and Mathematical Sciences

基　　金：山东省自然科学基金(ZR2022MA024);国家自然科学基金(12071264,11701336);山东省高等学校“青创人才引育计划”资助课题。

摘　　要：线性码的重量分布是编码理论的重要研究问题.少重量线性码在构造认证码、具有良好结构的秘密共享方案、结合方案、常量成份码、强正则图等有着广泛应用.设有限链环R=F_(q)+uF_(q),其中q是奇素数的幂次,F_(q)为q元有限域且u^(2)=0.文章利用有限链环R的两个扩环构造了R上的一类循环码,通过有限域上的高斯和理论给出了这类循环码的重量分布.作为应用,通过Gray映射构造了有限域上一类最优三重量线性码,并证明它是极小的.The weight distribution of linear codes is an important research problem in coding theory.Linear codes with a few weights are widely applied in constructing authentication codes,secret sharing schemes with nice access structures,association schemes,constant composition codes and strongly regular graphs.Let R=F_(q)+uF_(q) be a finite chain ring,where is an odd prime power,F_(q) denotes the finite field with elements and u^(2)=0.In this paper,we construct a family of cyclic codes over the finite chain ring,which are obtained from two extension rings of.By Gauss sums over finite fields,the weight distribution of this class of cyclic codes is investigated.As an application,a class of three-weight linear codes over finite fields which is optimal and minimal is constructed by the Gray map.

关 键 词：重量分布 高斯和 有限链环 最优线性码 

分 类 号：O157.4[理学—数学]