Rank-metric codes, linear sets, and their duality

TL;DR

This paper explores the relationship between linear sets, subspaces of linear maps, and their duality, providing geometric interpretations and extending results to higher dimensions, with applications to weight distributions in rank-metric codes.

Contribution

It introduces a geometric interpretation of linear sets, extends the connection between different constructions to arbitrary dimensions, and uses MacWilliams identities to analyze weight distributions.

Findings

Geometric interpretation of linear sets on projective lines

Extension of linear set constructions to higher dimensions

Determination of possible weight distributions using MacWilliams identities

Abstract

In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of [18, Section 5] on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in [9]. Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line $PG(1, q^n)$.