The subfield codes of hyperoval and conic codes

Abstract Hyperovals in PG ( 2 , GF ( q ) ) with even q are maximal arcs and an interesting research topic in finite geometries and combinatorics. Hyperovals in PG ( 2 , GF ( q ) ) are equivalent to [ q + 2 , 3 , q ] MDS codes over GF ( q ) , called hyperoval codes, in the sense that one can be constructed from the other. Ovals in PG ( 2 , GF ( q ) ) for odd q are equivalent to [ q + 1 , 3 , q − 1 ] MDS codes over GF ( 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.