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

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.