The Subfield Codes and Subfield Subcodes of a Family of MDS Codes

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, two infinite families of 3-designs from these quaternary codes and their duals are presented.