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

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.