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Recently, subfield codes of geometric codes over large finite fields $\gf(q)$with dimension $3$ and $4$ were studied and distance-optimal subfield codesover $\gf(p)$ were obtained, where $q=p^m$. The key idea for obtaining verygood subfield codes over small fields is to choose very good linear codes overan extension field with small dimension. This paper first presents a generalconstruction of $[q+1, 2, q]$ MDS codes over $\gf(q)$, and then studies thesubfield codes over $\gf(p)$ of some of the $[q+1, 2,q]$ MDS codes over$\gf(q)$. Two families of dimension-optimal codes over $\gf(p)$ are obtained,and several families of nearly optimal codes over $\gf(p)$ are produced.Several open problems are also proposed in this paper.

The Subfield Codes of $[q+1, 2, q]$ MDS Codes