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.
翻译：极正正正正方形草原代码$C(\ mathbb{O ⁇ 3,6}) $C(\\ mathb{O ⁇ 3,6}) 是格拉斯曼嵌入双极空间的线性代码 $Q( 5, q) 。 在此手稿中, 我们研究了嵌入的最小距离 。 我们证明极正方形草根代码$C( mathbb{O ⁇ 3,6}) 的最低距离 $C( mathb{O, 3,6}) $ 3- q 3$ 。 我们的技术以将正方形空间分割成不同的组合为基础, 这样, 每一个分区的代码 $C(\\\ mathb{O ⁇ 3, 6} 。 我们的边框来自计算某类多球的零值的基本等值。 我们期望我们的技术可以应用到其他极地偏偏偏差的代码 。
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