Linear Cutting Blocking Sets and Minimal Codes in the Rank Metric

TL;DR

This paper explores the structure of rank-metric codes using finite geometry concepts, introduces minimal rank-metric codes, and provides bounds, existence results, and explicit constructions with applications to error correction.

Contribution

It introduces and studies minimal rank-metric codes, providing bounds, existence results, and explicit constructions using scattered linear sets.

Findings

Bounds for parameters of minimal rank-metric codes

A general combinatorial existence result

Explicit code constructions using scattered linear sets

Abstract

This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the $q$-analogues of projective systems and blocking sets. We also illustrate how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence. The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles. Our main contributions are bounds for the parameters of a minimal rank-metric codes, a general existence result based on a combinatorial argument, and an explicit code construction for some parameter sets that uses the notion of a scattered linear set. Throughout the paper we also show and comment on curious analogies/divergences between the theories of error-correcting codes in the rank and in the Hamming metric.