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On good
infinite families of toric codes or the lack thereof
Mallory Dolorfino, Cordelia Horch, Kelly Jabbusch and Ryan Martinez |
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Vol. 17 (2024), No. 3, 397–423
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Abstract |
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Toric codes, introduced by Hansen, are the natural extensions of Reed–Solomon codes. A toric code is a -dimensional subspace of determined by a toric variety or its associated integral convex polytope , where is equal to (the number of integer lattice points of ). 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. |
Keywords
toric varieties, evaluation codes, toric codes, lattice
polytopes
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Mathematical Subject Classification
Primary: 94B27
Secondary: 14M25, 52B20
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Milestones
Received: 2 February 2022
Revised: 22 February 2023
Accepted: 27 February 2023
Published: 17 July 2024
Communicated by Ravi Vakil
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Authors |
| Mallory Dolorfino | |
| Kalamazoo College Kalamazoo, MI United States |
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| Cordelia Horch | |
| Occidental College Los Angeles, CA United States |
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| Kelly Jabbusch | |
| Department of Mathematics and
Statistics University of Michigan Dearborn, MI United States |
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| Ryan Martinez | |
| Harvey Mudd College Claremont, CA United States |
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| © 2024 MSP (Mathematical Sciences Publishers). |
