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arXiv · 1502.05808

Matrix Codes as Ideals for Grassmannian Codes and their Weight Properties

Bryan Hernandez·Virgilio Sison
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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.

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Bryan Hernandez, Virgilio Sison. 2015-02-20. Matrix Codes as Ideals for Grassmannian Codes and their Weight Properties. https://arxiv.org/abs/1502.05808

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