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arXiv · 2210.12884

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

Sarah Gregory·Fernando Piñero-González·Doel Rivera-Laboy·Lani Southern
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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.

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Sarah Gregory, Fernando Piñero-González, Doel Rivera-Laboy, Lani Southern. 2022-10-23. Computing the minimum distance of the $C(\mathbb{O}_{3,6})$ polar Orthogonal Grassmann code with elementary methods. https://arxiv.org/abs/2210.12884

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