Recently, subfield codes of geometric codes over large finite fieldsGF(q)with dimension 3 and 4 were studied and distance-optimal subfield codes overGF(p)were obtained, whereq=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 overGF(q), and then studies the subfield codes overGF(p)of some of the$[q+1, 2,q]$MDS codes overGF(q). Two families of dimension-optimal codes overGF(p)are obtained, and several families of nearly optimal codes overGF(p)are produced. Several open problems are also proposed in this paper.
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Heng et al. (2022) studied this question.
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