Computing the minimum distance of the $C(\mathbb{O}_{3,6})$ polar Orthogonal Grassmann code with elementary methods
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.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2022
- DOI:
- 10.48550/arXiv.2210.12884
- arXiv:
- arXiv:2210.12884
- Bibcode:
- 2022arXiv221012884G
- Keywords:
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- Mathematics - Combinatorics;
- Computer Science - Information Theory;
- Mathematics - Algebraic Geometry