Optimal anticodes, MSRD codes, and generalized weights in the sum-rank metric

TL;DR

This paper extends bounds and classifications for sum-rank metric codes, introduces generalized weights, and explores their properties and applications in network coding, advancing the theoretical understanding of these codes.

Contribution

It provides a new Anticode Bound for the sum-rank metric, classifies optimal anticodes, and analyzes generalized sum-rank weights, including their relation to MSRD codes.

Findings

Extended Anticode Bound for sum-rank metric

Classified optimal anticodes attaining the bound

Determined generalized weights of MSRD codes

Abstract

Sum-rank metric codes have recently attracted the attention of many researchers, due to their relevance in several applications. Mathematically, the sum-rank metric is a natural generalization of both the Hamming metric and the rank metric. In this paper, we provide an Anticode Bound for the sum-rank metric, which extends the corresponding Hamming and rank-metric Anticode bounds. We classify then optimal anticodes, i.e., codes attaining the sum-rank metric Anticode Bound. We use these optimal anticodes to define generalized sum-rank weights and we study their main properties. In particular, we prove that the generalized weights of an MSRD code are determined by its parameters. As an application, in the Appendix we explain how generalized weights measure information leakage in multishot network coding.