  Recently, subfield codes of geometric codes over large finite fields $\gf(q)$
with dimension $3$ and $4$ were studied and distance-optimal subfield codes
over $\gf(p)$ 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 $\gf(q)$, and then studies the
subfield 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.
