Eigenvalue bounds and alternating rank-metric codes

TL;DR

This work studies the problem of bounding the size of alternating rank-metric codes through a spectral lens, focusing on the graph of alternating bilinear forms. By analyzing the $\Gamma(\mathrm{Alt}_n(\mathbb{F}_q))$ graph, with eigenvalues given by $P^{(n)}(x)$, the authors derive ratio-type (Hoffman-like) eigenvalue bounds that bound the $k$-independence numbers and hence $A_q(n,2d)$. They show that for small minimum distances $2d$, these spectral bounds coincide with Delsarte's linear programming bounds (Singleton-like bound), and hence provide an easier computational route. Coding-theoretic bounds such as Code-Anticode, Sphere Packing, and Total Distance are surveyed and found to be weaker than the Singleton-like bound except in a few cases; the results suggest the spectral method can match LP bounds efficiently and may extend to larger $d$ with further work toward proving full equivalence. Overall, the paper highlights a practical, spectrum-based alternative to LP for bounding alternating rank-metric codes and opens questions about extending equivalence to all $d$ and bounding higher-order independence numbers like $\alpha_4$.

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.

Theorems & Definitions (53)

• Definition 2.1
• Theorem 2.2: delsarte1975alternating
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 3.1
• Lemma 3.2
• Lemma 3.3