The Subfield Codes of Hyperoval and Conic codes
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Hyperovals in (2,(q)) with even q are maximal arcs and an interesting research topic in finite geometries and combinatorics. Hyperovals in (2,(q)) are equivalent to [q+2,3,q] MDS codes over (q), called hyperoval codes, in the sense that one can be constructed from the other. Ovals in (2,(q)) for odd q are equivalent to [q+1,3,q-1] MDS codes over (q), which are called oval codes. In this paper, we investigate the binary subfield codes of two families of hyperoval codes and the p-ary subfield codes of the conic codes. The weight distributions of these subfield codes and the parameters of their duals are determined. As a byproduct, we generalize one family of the binary subfield codes to the p-ary case and obtain its weight distribution. The codes presented in this paper are optimal or almost optimal in many cases. In addition, the parameters of these binary codes and p-ary codes seem new.
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