超椭圆和圆锥码的子域码
The Subfield Codes of Hyperoval and Conic codes
摘要：
具有偶数$q$的$\PG(2,\gf(q))$中的超椭圆是极大弧，是有限几何和组合数学中一个有趣的研究课题。$\PG(2,\gf(q))$中的超椭圆等价于$\gf(q)$上的$[q+2,3,q]$ MDS码，称为超椭圆码，因为一个超椭圆码可以由另一个超椭圆码构造。奇数$q$的$\PG(2,\gf(q))$中的椭圆等价于$\gf(q)$上的$[q+1,3,q-1]$ MDS码，称为椭圆码。本文研究了两类超椭圆码的二元子域码和圆锥码的$p$元子域码。确定了这些子字段码的权值分布和子字段码的参数。作为副产品，我们推广了一个家庭的二元子域码的$p$ -ary的情况下，并获得其重量分布。本文提出的码在许多情况下是最优或几乎最优的。此外，这些二进制码和$p$ -进制码的参数似乎是新的。
英文摘要：
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.
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