The Subfield Codes of [q+1, 2, q] MDS Codes
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Recently, subfield codes of geometric codes over large finite fields (q) with dimension 3 and 4 were studied and distance-optimal subfield codes over (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 (q), and then studies the subfield codes over (p) of some of the [q+1, 2,q] MDS codes over (q). Two families of dimension-optimal codes over (p) are obtained, and several families of nearly optimal codes over (p) are produced. Several open problems are also proposed in this paper.
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