Delsarte duality on subspaces and applications to rank-metric codes and q-matroids

Abstract

We study the interplay between the lattice of F_{q^m}-subspaces and the lattice of F_{q^m}-subspaces of an F_{q^m}-vector space. Introducing notions of weight and defect relative to an F_q-subspace, we analyze the sequence of maximum non-zero defects. We establish a correspondence between subspaces of positive defect and their Delsarte duals, enabling explicit characterizations of the associated sequences of maximum non-zero defects. Our framework unifies several classes of subspaces studied in finite geometry and connects them to linear rank-metric codes by providing a new geometric interpretation of code duality. Building on these results, we characterize classes of rank-metric codes closed under duality, including MRD, near MRD, quasi-MRD, and a new family of (n, k)-MRD codes. Finally, we explore applications to q-matroids, by studying the problem of F_{q^m}-representability for direct sums of uniform q-matroids and describing their rank generating functions.