Grassmannian Codes as Lifts of Matrix Codes Derived as Images of Linear Block Codes over Finite Fields

TL;DR

This paper constructs and analyzes Grassmannian codes as lifts of matrix codes derived from linear block codes over finite fields, demonstrating their optimality and providing new examples of maximum rank distance and anticode-optimal Grassmannian codes.

Contribution

It introduces a novel method to derive Grassmannian codes from matrix codes over finite fields, establishing their optimality and providing new explicit examples.

Findings

Matrix codes are isometric images of codes over GF(p^2).

Derived codes satisfy the Singleton bound as maximum rank distance codes.

Constructed Grassmannian codes meet the anticode bound and are associated with complete graphs.

Abstract

Let $p$ be a prime such that $p \equiv 2$ or $3$ mod $5$. Linear block codes over the non-commutative matrix ring of $2 \times 2$ matrices over the prime field $GF(p)$ endowed with the Bachoc weight are derived as isometric images of linear block codes over the Galois field $GF(p^2)$ endowed with the Hamming metric. When seen as rank metric codes, this family of matrix codes satisfies the Singleton bound and thus are maximum rank distance codes, which are then lifted to form a special class of subspace codes, the Grassmannian codes, that meet the anticode bound. These so-called anticode-optimal Grassmannian codes are associated in some way with complete graphs. New examples of these maximum rank distance codes and anticode-optimal Grassmannian codes are given.