Constant-Rank Codes

TL;DR

This paper explores the properties of constant-rank codes, their relation to constant-dimension codes, and derives bounds and asymptotic behavior for their maximum size, contributing to error control in network coding.

Contribution

It establishes a connection between constant-rank and constant-dimension codes and derives bounds and asymptotic results for constant-rank codes.

Findings

Relation between vectors in GF(q^m)^n and subspaces of GF(q)^m or GF(q)^n

Bounds on maximum cardinality of constant-rank codes

Asymptotic behavior of maximal cardinality of constant-rank codes

Abstract

Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random network coding. In this paper, we show that constant-rank codes are closely related to constant-dimension codes and we study the properties of constant-rank codes. We first introduce a relation between vectors in $\mathrm{GF}(q^m)^n$ and subspaces of $\mathrm{GF}(q)^m$ or $\mathrm{GF}(q)^n$, and use it to establish a relation between constant-rank codes and constant-dimension codes. We then derive bounds on the maximum cardinality of constant-rank codes with given rank weight and minimum rank distance. Finally, we investigate the asymptotic behavior of the maximal cardinality of constant-rank codes with given rank weight and minimum rank distance.