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

Chunming Tang, Qi Wang, Cunsheng Ding

Abstract: Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[{2^{m}+1, 2u-1, 2^{m}-2u+3}]$ </tex-math></inline-formula> MDS codes for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1 \leq u \leq 2^{m-1}$ </tex-math></inline-formula> , which are cyclic, reversible and BCH codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(2^{m})$ </tex-math></inline-formula> . 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> . 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.