Matrix Codes as Ideals for Grassmannian Codes and their Weight Properties

TL;DR

This paper introduces a systematic method for constructing Grassmannian codes from matrix codes over finite fields, analyzing their weight properties and distributions using algebraic structures like ideals and group actions.

Contribution

It presents a novel approach to constructing Grassmannian codes via matrix ideals and studies their weight functions and distributions in a non-commutative setting.

Findings

Complete rank weight distribution of $M_2(GF(q))$ determined by $GL(2,q)

Weight functions on matrix rings exhibit egalitarian and homogeneous properties

New weight function defined on subspace codes with examined properties

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 $GL(2,q)$. Finally a weight function on subspace codes is analogously defined and its egalitarian property is examined.