The subfield codes and subfield subcodes of a family of MDS codes
(2106.07840)Abstract
Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[2m+1, 2u-1, 2m-2u+3]$ MDS codes for $1 \leq u \leq 2{m-1}$, which are cyclic, reversible and BCH codes over $\mathrm{GF}(2m)$. 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.
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