Matrix Codes as Ideals for Grassmannian Codes and their Weight Properties

Bryan Hernandez; Virgilio Sison

paper

Matrix Codes as Ideals for Grassmannian Codes and their Weight Properties

arXiv:1502.05808

Abstract

A systematic way of constructing Grassmannian codes endowed with the subspace distance as lifts of matrix codes over the prime field $GF (p)$ is introduced. The matrix codes are $GF (p)$ -subspaces of the ring $M_{2} (GF (p))$ of $2 \times 2$ matrices over $GF (p)$ on which the rank metric is applied, and are generated as one-sided proper principal ideals by idempotent elements of $M_{2} (GF (p))$ . Furthermore a weight function on the non-commutative matrix ring $M_{2} (GF (p))$ , $q$ a power of $p$ , is studied in terms of the egalitarian and homogeneous conditions. The rank weight distribution of $M_{2} (GF (q))$ is completely determined by the general linear group $G L (2, q)$ . Finally a weight function on subspace codes is analogously defined and its egalitarian property is examined.

References in corpus (1)

Projective Space Codes for the Injection Metric