Optimal Anticodes, MSRD Codes, and Generalized Weights in the Sum-Rank Metric

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.

Automated summary

This paper introduces the Anticode Bound for sum-rank metric codes, classifies optimal anticodes, defines generalized sum-rank weights, and studies their main properties. It proves that the generalized weights of an MSRD code are determined by its parameters. Additionally, the paper explains how generalized weights measure information leakage in multishot network coding in the Appendix.