Linear Cutting Blocking Sets and Minimal Codes in the Rank Metric

Gianira Nicoletta Alfarano, Martino Borello, Alessandro Neri, and Alberto Ravagnani

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Submission date: 11. Feb. 2022
Bibtex
Keywords and phrases: rank-metric codes, minimal codes, linear sets, linear cutting blocking sets, blocking sets
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the q-analogues of projective systems and blocking sets. We also illustrate how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence. The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles. Our main contributions are bounds for the parameters of a minimal rank-metric codes, a general existence result based on a combinatorial argument, and an explicit code construction for some parameter sets that uses the notion of a scattered linear set. Throughout the paper we also show and comment on curious analogies/divergences between the theories of error-correcting codes in the rank and in the Hamming metric.