Linear codes associated with the Desarguesian ovoids in $Q^+(7,q)$

TL;DR

This paper explores the structure of linear codes derived from Desarguesian ovoids in a specific orthogonal polar space, providing parameters, weight distributions, and optimality results for these codes across all prime power values of q.

Contribution

It determines the parameters and weight distributions of codes associated with Desarguesian ovoids in $Q^+(7,q)$ for all prime powers q, and proves their length optimality.

Findings

Parameters of the codes $C_O$ and $C_O^ot$ are established.

Weight distributions of the codes are computed.

Codes are shown to be length-optimal for all prime powers q.

Abstract

The Desarguesian ovoids in the orthogonal polar space $Q^+(7,q)$ with $q$ even have first been introduced by Kantor by examining the $8$-dimensional absolutely irreducible modular representations of $\text{PGL}(2,q^3)$. We investigate this module for all prime power values of $q$. The shortest $\text{PGL}(2,q^3)$-orbit $O$ gives the Desarguesian ovoid in $Q^+(7,q)$ for even $q$ and it is known to give a complete partial ovoid of the symplectic polar space $W(7,q)$ for odd~$q$. We determine the hyperplane sections of $O$. As a corollary, we obtain the parameters $[q^3+1,8,q^3-q^2-q]_q$ and the weight distribution of the associated $\mathbb{F}_q$-linear code $C_O$ and the parameters $[q^3+1,q^3-7,5]_q$ of the dual code $C_O^\perp$ for $q \ge 4$. We also show that both codes $C_O$ and $C_O^\perp$ are length-optimal for all prime power values of $q$.