Rank-metric codes as ideals for subspace codes and their weight properties

Abstract

Let q=pr, p a prime, r a positive integer, and Fq the Galois field with cardinality q and characteristic p. In this paper, we study some weight properties of rank-metric codes and subspace codes. The rank weight is not egalitarian nor homogeneous, and the rank weight distribution of M2(Fq) is completely determined by the general linear group GL(2,q). We consider subspace weight that is defined on subspace codes and examine their egalitarian property. We also present some examples of rank-metric codes endowed with the rank distance and Grassmannian codes endowed with the subspace distance. These codes were generated from left ideals of M2(Fp) using idempotent elements of M2(Fp).