渐近好的加性循环码的存在性  

Asymptotically Good Additive Cyclic Codes Exist

作　　者：姚婷 唐永生 许和乾 YAO Ting;TANG Yongsheng;XU Heqian(School of Mathematics and Statistics,Hefei Normal University,Hefei 230601)

机构地区：[1]合肥师范学院数学与统计学院,合肥230601

出　　处：《系统科学与数学》2024年第12期3779-3789,共11页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金(12201170,12171134,62172183);安徽省自然科学基金(2108085QA03);合肥师范学院2022年度引进高层次人才科研启动基金(2022rcjj11);安徽省高等学校自然科学研究项目(KJ2021A0926)资助课题。

摘　　要：文章研究了由三元组多项式生成的一类Z_(p^(r)),Z_(p^(s))Z_(p^(t-))加性循环码,其中p是一个素数,1≤r≤s≤t.通过建立随机Z_(p^(r)),Z_(p^(s))Z_(p^(t-))加性循环码与指数为3的准阿贝尔互补对偶码的联系,文章确定了该类加性循环码的相对最小距离收敛于δ,码率收敛于1/3其中0<δ<3/2(1+p^(s-r)+p^(t-r)).最后,文章证明了存在大量渐近好的Z_(p^(r)),Z_(p^(s))Z_(p^(t-))加性循环码.A class of ZprZps Zpt-additive cyclic codes that are formed by 3-tuples of polynomials are constructed in this paper,where p is a prime number and 1≤r≤s≤t.Having established a connection between the random Z_(p)^(r) Z_(p)^(s) Z_(p)^(t)-additive cyclic codes and the quasi-abelian complementary dual codes of index 3,we construct a family of codes of increasing length whose relative minimum distances converge d while their rates converge1/3 ,for 0<δ<3/2(1+p^(s-r)+p^(t-r)).Consequentywedemonstrate that a large number of Z_(p)^(r) Z_(p)^(s) Z_(p)^(t)-additive cyclic codes are asymptotically good.

关 键 词：Z_(p^(r))Z_(p^(s))Z_(p^(t^(-)))加性循环码 随机码 模-同构 相对距离 渐近好码 

分 类 号：O157.4[理学—数学]