A geometric characterization of minimal codes and their asymptotic performance

Gianira N. Alfarano; Martino Borello; Alessandro Neri

doi:10.3934/amc.2020104

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2022, Volume 16, Issue 1: 115-133. Doi: 10.3934/amc.2020104

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A geometric characterization of minimal codes and their asymptotic performance

1.
Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
2.
Université Paris 8, Laboratoire de Géométrie, Analyse et Applications, LAGA, Université Sorbonne Paris Nord, CNRS, UMR 7539, F-93430, Villetaneuse, France
3.
Institute for Communications Engineering, Technical University of Munich, Theresienstrasse 90, 80333 Munich, Germany

^* Corresponding author: Alessandro Neri
^* Corresponding author: Alessandro Neri

Received: December 31, 2019

Revised: May 31, 2020

Early access: August 2020

Published: February 2022

G. N. Alfarano acknowledges the support of Swiss National Science Foundation grant n. 188430. A. Neri acknowledges the support of Swiss National Science Foundation grant n. 187711

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Abstract

In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive some bounds on the length and the distance of minimal codes, according to their dimension and the underlying field size. Furthermore, we show that the family of minimal codes is asymptotically good. Finally, we provide some geometrical constructions of minimal codes as cutting blocking sets.

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Mathematics Subject Classification: 51E21, 51E20, 94B05.

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