An algebraic-coding equivalence to the maximum distance separable conjecture

Damelin, Steven; Kaiser, Daniel; Sun, Jeffrey; Bora, Safal

doi:10.2140/involve.2024.17.363

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An algebraic-coding equivalence to the maximum distance separable conjecture

Steven Damelin, Daniel Kaiser, Jeffrey Sun and Safal Bora

Vol. 17 (2024), No. 3, 363–372

DOI: 10.2140/involve.2024.17.363

Abstract

Let $k$ be an integer such that $2 \leq k \leq q = p^{r}$ , where $p$ is prime and $r$ is a positive integer. A $k \times n$ maximum distance separable ( $k \times n$ MDS) code $M$ is a $k \times n$ matrix with entries in $𝔽_{q}$ such that every set of $k$ columns of $M$ is linearly independent.

The maximum distance separable (MDS) conjecture is a well-known conjecture in coding theory and algebraic geometry with important consequences, for example, to the study of arcs in finite projective spaces and to coding theory. The conjecture is the following: the maximum width, $n$ , of a $k \times n$ MDS code with entries in $𝔽_{q}$ is $q + 1$ , unless $q$ is even and $k \in {3, q - 1}$ , in which case the maximum width is $q + 2$ . We give necessary and sufficient conditions for the MDS conjecture to hold.

Keywords

maximum distance separable conjecture, linear independence, coding theory, algebraic geometry, projective spaces, arcs

Mathematical Subject Classification 2010

Primary: 94B05

Secondary: 05B35, 12Y05

Milestones

Received: 14 October 2018

Revised: 1 March 2023

Accepted: 2 March 2023

Published: 17 July 2024

Communicated by Ken Ono

Authors

Steven Damelin
Department of Mathematics University of Michigan Ann Arbor, MI United States

Daniel Kaiser
Department of Mathematics University of Michigan Ann Arbor, MI United States

Jeffrey Sun
Department of Mathematics University of Michigan Ann Arbor, MI United States

Safal Bora
Department of Mathematics University of Michigan Ann Arbor, MI United States

Vol. 17, No. 3, 2024