Matrix Codes as Ideals for Grassmannian Codes and their Weight Properties

Abstract—A systematic way of constructing Grassmannian codes endowed with the subspace distance as lifts of matrix codes over the prime field Fp is introduced. The matrix codes are Fp-subspaces of the ring M2(Fp) of 2 × 2 matrices over Fp on which the rank metric is applied, and are generated as onesided proper principal ideals by idempotent elements of M2(Fp). Furthermore a weight function on the non-commutative matrix ring M2(Fq), q a power of p, is studied in terms of the egalitarian and homogeneous conditions. The rank weight distribution of M2(Fq) is completely determined by the general linear group GL(2, q). Finally a weight function on subspace codes is analogously defined and its egalitarian property is examined.