The Subfield Codes of Some [q + 1, 2, q] MDS Codes
Ziling Heng, Cunsheng Ding · IEEE Transactions on Information Theory · 2022
Recently, subfield codes of geometric codes over large finite fields${\mathrm {GF}}(q)$with dimension 3 and 4 were studied and distance-optimal subfield codes over${\mathrm {GF}}(p)$were obtained, where$q=p^{m}$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of$[q+1, 2, q]$MDS codes over${\mathrm {GF}}(q)$, and then studies the subfield codes over${\mathrm {GF}}(p)$of some of the$[q+1, 2,q]$MDS codes over${\mathrm {GF}}(q)$. Two families of dimension-optimal codes over${\mathrm {GF}}(p)$are obtained, and several families of nearly optimal codes over${\mathrm {GF}}(p)$are produced. Several open problems are also proposed in this paper.