Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[2^m+1, 2u-1, 2^m-2u+3]$ MDS codes for $1 \leq u \leq 2^{m-1}$, which are cyclic, reversible and BCH codes over $\mathrm{GF}(2^m)$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, infinite families of $3$-designs from these quaternary codes are presented.
翻译：在理论和实践上,最大距离代码(MDS)在理论和实践上都非常重要。 典型地构建了$[2+1, 2u-1, 2 ⁇ -2u+3] 家庭代码($1\leq u\leq 2 ⁇ m-1}$),这些代码循环、可逆和BCH代码超过$\mathrm{GF}($2 ⁇ m)。本文的目的是研究MDS 代码下组的四硝基亚代码和四硝基亚字段代码($甚至为$)。 获得了四硝基周期代码的组合。 这些四甲代代码在某些情况下是距离最优的,而且总的来说非常良好。 此外,还介绍了这些四甲基代码中三美元的无限序列。
相关内容
微信扫码咨询专知VIP会员