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 family $[{2^{m}+1, 2u-1, 2^{m}-2u+3}]$ MDS for notation="LaTeX">$1 \leq u 2^{m-1}$ , which cyclic, reversible BCH over notation="LaTeX">${\mathrm {GF}}(2^{m})$ . The objective this paper to study the quaternary subfield subcodes subfamily even notation="LaTeX">$m$ A cyclic obtained. These distance-optimal some cases good general. Furthermore, two infinite families 3-designs from these their duals presented.