Constant-Rank Codes and Their Connection to Constant-Dimension Codes

TL;DR

This paper establishes a connection between constant-rank and constant-dimension codes, providing bounds and constructions for optimal constant-rank codes to solve key problems in network coding.

Contribution

It introduces a novel approach linking constant-rank and constant-dimension codes, and offers bounds and explicit constructions for optimal constant-rank codes.

Findings

Derived bounds on maximum cardinality of constant-rank codes

Proposed explicit constructions of optimal constant-rank codes

Established asymptotic bounds on the rate of constant-rank codes

Abstract

Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random linear network coding. What the maximal cardinality of any constant-dimension code with finite dimension and minimum distance is and how to construct the optimal constant-dimension code (or codes) that achieves the maximal cardinality both remain open research problems. In this paper, we introduce a new approach to solving these two problems. We first establish a connection between constant-rank codes and constant-dimension codes. Via this connection, we show that optimal constant-dimension codes correspond to optimal constant-rank codes over matrices with sufficiently many rows. As such, the two aforementioned problems are equivalent to determining the maximum cardinality of constant-rank codes and to constructing optimal constant-rank codes, respectively. To this end, we then derive bounds on the maximum cardinality of a constant-rank code with a given minimum rank distance, propose explicit constructions of optimal or asymptotically optimal constant-rank codes, and establish asymptotic bounds on the maximum rate of a constant-rank code.