This article is cited in 3 papers

Coding Theory

Non-split toric codes

D. I. Koshelev<sup>abc

a</sup> Department of Discrete Mathematics, Moscow Institute of Physics and Technology (State University), Moscow, Russia
^b Versailles Laboratory of Mathematics, Versailles Saint-Quentin-en-Yvelines University, Versailles, France
^c Algebra and Number Theory Laboratory, Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

Abstract: We introduce a new wide class of error-correcting codes, called non-split toric codes. These codes are a natural generalization of toric codes where non-split algebraic tori are taken instead of usual (i.e., split) ones. The main advantage of the new codes is their cyclicity; hence, they can possibly be decoded quite fast. Many classical codes, such as (doubly-extended) Reed–Solomon and (projective) Reed–Muller codes, are contained (up to equivalence) in the new class. Our codes are explicitly described in terms of algebraic and toric geometries over finite fields; therefore, they can easily be constructed in practice. Finally, we obtain new cyclic reversible codes, namely non-split toric codes on the del Pezzo surface of degree $6$ and Picard number $1$. We also compute their parameters, which prove to attain current lower bounds at least for small finite fields.

Keywords: finite fields, toric and cyclic codes, non-split algebraic tori, toric varieties, del Pezzo surfaces, elliptic curves.

UDC: 621.391.15

Received: 22.11.2018
Revised: 09.01.2019
Accepted: 15.01.2019

DOI: 10.1134/S0555292319020025

 English version: Problems of Information Transmission, 2019, 55:2, 124–144

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