有限域上码长为7p^(s)的重根循环码  

Repeated-Root Cyclic Codes of Length 7p^(s) over Finite Fields

作　　者：黄霞 李锦[1] 黄山 HUANG Xia;LI Jin;HUANG Shan(School of Mathematics,Hefei University of Technology,Hefei 230601;Department of Information Management,Anhui Vocational College of Police Officers,Hefei 230031)

机构地区：[1]合肥工业大学数学学院,合肥230601 [2]安徽警官职业学院信息管理系,合肥230031

出　　处：《系统科学与数学》2022年第3期766-778,共13页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金(62002093);中央高校基本科研业务费专项资金(PA2019GDZC0097)资助课题。

摘　　要：线性码是一类非常重要的纠错码,线性码的最小汉明距离决定了其检错纠错能力,然而如何确定线性码的最小汉明距离至今仍是一难题.循环码是线性码的一个重要子类,已有广泛的应用.文章研究了有限域F_(q)上码长为7p^(s)的重根循环码的最小汉明距离,其中q=p^(m),p≥11为素数,m,s为正整数.先确定了有限域F_(q)上所有码长为7的单根循环码的最小汉明距离,进而确定了码长为7p^(s)的重根循环码的最小汉明距离,并得到了一些达到Griesmer界的最优重根循环码,最后利用码长为7p^(s)的对偶包含重根循环码构造了量子同步码.Linear codes are a very significant class of error correcting codes.The minimum Hammin g distances of linear codes determine their ability of detecting errors and correcting errors.However,how to determine the minimum Hamming distances of linear codes is still a difficult problem.Cyclic codes are an important subclass of linear codes and have been widely used.In this paper,we study the minimum Hamming distances of repeated-root cyclic codes of length 7p^(s)over the finite field F_(q),where q=p^(m),p≥11 is a prime,m,s are positive integers.Using the minimum Hamming distances of cyclic codes of length 7 over F_(q),we determine the minimum Hamming distances of repeated-root cyclic codes of length 7p^(s)over F_(q).And we obtain some optimal repeated-root cyclic codes of length 7p^(s)with respect to the Griesmer bound.As an application,we construct quantum synchronizable codes from dual-containing repeated-root cyclic codes.

关 键 词：重根循环码 最小汉明距离 Griesmer界 量子同步码 

分 类 号：O157.4[理学—数学]