2022/10/23 by Gregory, Sarah, Piñero-González, Fernando, Rivera-Laboy, Doel +1
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2210.12884
The polar orthogonal Grassmann code C(\mathbbO3,6) 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(\mathbbO3,6) is q3-q3 for q odd and q3 for q even. Our technique is based on partitioning the orthogonal space into different sets such that on each partition the code C(\mathbbO3,6) 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.