Computing the minimum distance of the C(O3,6) polar orthogonal Grassmann code with elementary methods

The polar orthogonal Grassmann code $C\left(O_{3,6}\right)$ is the linear code associated to the polar orthogonal Grassmannian subvariety of the Grassmannian. The variety $O_{3,6}$ is the Grassmannian of 3-spaces contained in a hyperbolic quadric in $PG(6,q)$. In this manuscript we prove that the minimum distance of the polar orthogonal Grassmann code $C\left(O_{3,6}\right)$ is $q^3-q^2$ for q odd and $q^3$ for q even. We also prove that the minimum distance of $C\left(O_{4,8}\right)$ is $q^6$ when q is even. Our technique is based on partitioning $O_{3,6}$ into different sets such that on each partition the code $C\left(O_{3,6}\right)$ 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. The techniques presented in this paper may be adapted for other polar Grassmannians.