Abstract
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 = pm. 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 dimensionoptimal 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.
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