On MSRD codes, h-designs and disjoint maximum scattered linear sets

TL;DR

This paper explores the geometric properties of sum-rank metric codes, establishing new connections with h-designs and constructing novel MSRD codes through disjoint maximum scattered linear sets.

Contribution

It introduces a geometric framework for sum-rank codes, extending known correspondences and providing new constructions of MSRD codes and h-designs.

Findings

Established geometric description of generalized weights

Extended correspondence between MSRD codes and h-designs

Constructed new MSRD codes using disjoint maximum scattered linear sets

Abstract

In this paper we study geometric aspects of codes in the sum-rank metric. We establish the geometric description of generalised weights, and analyse the Delsarte and geometric dual operations. We establish a correspondence between maximum sum-rank distance codes and h-designs, extending the well-known correspondence between MDS codes and arcs in projective spaces and between MRD codes and h-scatttered subspaces. We use the geometric setting to construct new h-designs and new MSRD codes via new families of pairwise disjoint maximum scattered linear sets.