Eigenvalue bounds and alternating rank-metric codes

TL;DR

This paper uses spectral methods on graphs of alternating bilinear forms to derive bounds on the size of rank-metric codes, showing equivalence with Delsarte's bounds for small distances and highlighting open problems for larger distances.

Contribution

It introduces a spectral approach to bound the size of alternating rank-metric codes and demonstrates their equivalence with existing linear programming bounds for small minimum distances.

Findings

Spectral bounds match Delsarte's linear programming bounds for small distances

Established equivalence of methods for small minimum rank distances

Open problem remains for larger minimum distances

Abstract

In this note we apply a spectral method to the graph of alternating bilinear forms. In this way, we obtain upper bounds on the size of an alternating rank-metric code for given values of the minimum rank distance. We computationally compare our results with Delsarte's linear programming bound, observing that they give the same value. For small values of the minimum rank distance, we are able to establish the equivalence of the two methods. The problem remains open for larger values.