Extended cyclic codes, maximal arcs and ovoids

TL;DR

This paper explores the connection between extended cyclic codes over finite fields and geometric structures like hyperovals, maximal arcs, and ovoids, providing new characterizations and presentations of these objects.

Contribution

It establishes new links between specific classes of extended cyclic codes and geometric configurations such as hyperovals, maximal arcs, and ovoids, offering novel characterizations and simpler presentations.

Findings

Extended cyclic codes with parameters [q+2,3,q] determine regular hyperovals.

Codes with parameters [qt-q+t,3,qt-q] determine Denniston maximal arcs.

Codes with parameters [q^2+1,4,q^2-q] are equivalent to ovoid codes from elliptic quadrics.

Abstract

We show that extended cyclic codes over $\mathbb{F}_q$ with parameters $[q+2,3,q]$, $q=2^m$, determine regular hyperovals. We also show that extended cyclic codes with parameters $[qt-q+t,3,qt-q]$, $1<t<q$, determine (cyclic) Denniston maximal arcs. Similarly, cyclic codes with parameters $[q^2+1,4,q^2-q]$ are equivalent to ovoid codes obtained from elliptic quadrics in $PG(3,q)$. Finally, we give new simple presentations of Denniston maximal arcs in $PG(2,q)$ and elliptic quadrics in $PG(3,q)$.