The Subfield Codes of Hyperoval and Conic codes

TL;DR

This paper studies subfield codes derived from hyperoval and conic codes in finite projective planes, determining their weight distributions, parameters, and optimality, with some results being new and generalizations of existing codes.

Contribution

It introduces new subfield codes from hyperoval and conic codes, determines their weight distributions and parameters, and generalizes binary codes to the p-ary case.

Findings

Codes are optimal or nearly optimal in many cases.

Parameters of these codes are new and previously unreported.

Generalization of binary subfield codes to p-ary case achieved.

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.