论文标题
计算$ c（\ mathbb {o} _ {3,6}）的最小距离，$ polar正交Grassmann代码使用基本方法
Computing the minimum distance of the $C(\mathbb{O}_{3,6})$ polar Orthogonal Grassmann code with elementary methods
论文作者
论文摘要
极性正交Grassmann代码$ C（\ Mathbb {O} _ {3,6}）$是与$ Q^+（5，q）$的Grassmann嵌入的Grassmann嵌入相关的线性代码。在本手稿中，我们研究了此嵌入的最小距离。我们证明，极性正交的Grassmann代码$ C（\ Mathbb {o} _ {3,6}）$的最小距离是$ q^3-q^3 $ for $ q $奇数和$ q $奇数和$ q^3 $偶数。我们的技术基于将正交空间划分为不同的集合，以便在每个分区上$ c（\ mathbb {o} _ {3,6}）$确定，并通过评估skew--对称矩阵的确定因素进行评估。我们的边界来自计算特定类别多项式类别的零的基本代数方法。我们预计我们的技术可能会应用于其他极地格拉曼码。
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.
