Linear Cutting Blocking Sets and Minimal Codes in the Rank Metric
Abstract
This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the -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.
Cite
@article{arxiv.2106.12465,
title = {Linear Cutting Blocking Sets and Minimal Codes in the Rank Metric},
author = {Gianira N. Alfarano and Martino Borello and Alessandro Neri and Alberto Ravagnani},
journal= {arXiv preprint arXiv:2106.12465},
year = {2021}
}