On the covering dimension of a linear code

TL;DR

This paper introduces the covering dimension of linear codes, relating it to matroid theory's critical exponent, proposes an upper bound conjecture, and provides constructions achieving this bound.

Contribution

It defines the covering dimension for linear codes, proposes a nearly proven upper bound, and constructs codes that meet this bound, advancing understanding in coding theory and matroid connections.

Findings

Proposed a conjectured upper bound on covering dimension.

Established a near-proof of the upper bound.

Constructed codes that attain the bound.

Abstract

The critical exponent of a matroid is one of the important parameters in matroid theory and is related to the Rota and Crapo's Critical Problem. This paper introduces the covering dimension of a linear code over a finite field, which is analogous to the critical exponent of a representable matroid. An upper bound on the covering dimension is conjectured and nearly proven, improving a classical bound for the critical exponent. Finally, a construction is given of linear codes that attain equality in the covering dimension bound.