On good infinite families of toric codes or the lack thereof

Dolorfino, Mallory; Horch, Cordelia; Jabbusch, Kelly; Martinez, Ryan

doi:10.2140/involve.2024.17.397

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On good infinite families of toric codes or the lack thereof

Mallory Dolorfino, Cordelia Horch, Kelly Jabbusch and Ryan Martinez

Vol. 17 (2024), No. 3, 397–423

DOI: 10.2140/involve.2024.17.397

Abstract

Toric codes, introduced by Hansen, are the natural extensions of Reed–Solomon codes. A toric code is a $k$ -dimensional subspace of $𝔽_{q}^{n}$ determined by a toric variety or its associated integral convex polytope $P \subseteq {[0, q - 2]}^{n}$ , where $k$ is equal to $| P \cap ℤ^{n} |$ (the number of integer lattice points of $P$ ). There are two relevant parameters that determine the quality of a code: the information rate, which measures how much information is contained in a single bit of each codeword, and the relative minimum distance, which measures how many errors can be corrected relative to how many symbols each codeword has. Soprunov and Soprunova defined a good infinite family of codes to be a sequence of codes of unbounded polytope dimension such that neither the corresponding information rates nor relative minimum distances go to 0 in the limit. We examine different ways of constructing families of codes by considering polytope operations such as the join and direct sum. In doing so, we give conditions under which no good family can exist and strong evidence that there is no such good family of codes.

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Keywords

toric varieties, evaluation codes, toric codes, lattice polytopes

Mathematical Subject Classification

Primary: 94B27

Secondary: 14M25, 52B20

Milestones

Received: 2 February 2022

Revised: 22 February 2023

Accepted: 27 February 2023

Published: 17 July 2024

Communicated by Ravi Vakil

Authors

Mallory Dolorfino
Kalamazoo College Kalamazoo, MI United States

Cordelia Horch
Occidental College Los Angeles, CA United States

Kelly Jabbusch
Department of Mathematics and Statistics University of Michigan Dearborn, MI United States

Ryan Martinez
Harvey Mudd College Claremont, CA United States

Vol. 17, No. 3, 2024