Computing the minimum distance of the $C(\mathbb{O}_{3,6})$ polar Orthogonal Grassmann code with elementary methods

TL;DR

This paper determines the minimum distance of the polar orthogonal Grassmann code $C( ext{O}_{3,6})$ using elementary algebraic methods, distinguishing cases for odd and even q, and suggests broader applicability of these techniques.

Contribution

The authors compute the minimum distance of $C( ext{O}_{3,6})$ with elementary methods, providing explicit results for odd and even q, and introduce a novel approach based on partitioning orthogonal spaces.

Findings

Minimum distance is $q^3 - q^3$ for odd q.

Minimum distance is $q^3$ for even q.

Elementary algebraic methods effectively determine code parameters.

Abstract

The polar orthogonal Grassmann code $C(\mathbb{O}_{3,6})$ is the linear code associated to the Grassmann embedding of the Dual Polar space of $Q^+(5,q)$. In this manuscript we study the minimum distance of this embedding. We prove that the minimum distance of the polar orthogonal Grassmann code $C(\mathbb{O}_{3,6})$ is $q^3-q^3$ for $q$ odd and $q^3$ for $q$ even. Our technique is based on partitioning the orthogonal space into different sets such that on each partition the code $C(\mathbb{O}_{3,6})$ is identified with evaluations of determinants of skew--symmetric matrices. Our bounds come from elementary algebraic methods counting the zeroes of particular classes of polynomials. We expect our techniques may be applied to other polar Grassmann codes.