有限域上二次剩余码的两类推广  

Two Generalizations for Quadratic Residue Codes over Finite Fields

作　　者：廖群英 刘渊博 LIAO Qunying;LIU Yuanbo(School of Mathematical Sciences,Sichuan Normal University,Chengdu,Sichuan,610068,P.R.China)

机构地区：[1]四川师范大学数学科学学院,四川成都610068

出　　处：《数学进展》2021年第6期917-939,共23页Advances in Mathematics（China）

基　　金：Supported by NSFC(No.12071321)。

摘　　要：众所周知,有限域上二次剩余码是一类特殊的循环码,有较大的最小距离.设g是正整数,m是大于1的正整数且p,p_(1),p_(2),…,p_(g)为不同的奇质数.本文将有限域上码长为奇质数的二次剩余码的概念推广到码长为n=Πi=_(1)^(g) p_(i)的二次剩余码和码长为p的m次剩余码,得到这些码是LCD码或自正交码的充要条件,并给出相应的计数公式.特别地,本文确定了所有24个四元二次剩余码[15,8]及二元二次剩余码[161,81]的最小距离.It is well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance.Let g be a positive integer and p,p_(1),p_(2),…,p_(g) be distinct odd primes.The present paper generalizes the constructions for the quadratic residue code with length p to be the length n=Πi=_(1)^(g) p_(i),and to be the case m-th residue codes with length p over finite fields,where m≥2 is a positive integer.Furthermore,a criterion for that these codes are self-orthogonal or complementary dual is obtained,and then the corresponding counting formula is given.In particular,the minimum distances of all 24 quaternary quadratic residue codes[15,8]and all 24 binary quadratic residue codes[161,81]are determined.

关 键 词：二次剩余码 补对偶码 自正交码 m次剩余码 

分 类 号：O157.4[理学—数学]