Toric Codes and Finite Geometries

TL;DR

This paper explores how affine geometries over rings relate to the properties of toric codes over finite fields, providing new insights into their minimum distances and guiding the search for better codes.

Contribution

It introduces a geometric perspective using affine geometries over rings to analyze toric codes, connecting code properties to geometric structures and explaining their success over certain fields.

Findings

Minimum distance linked to lines in finite geometry

Geometric visualization aids in code analysis

Insights explain success over field F_8

Abstract

We show how the theory of affine geometries over the ring ${\mathbb Z}/\langle q - 1\rangle$ can be used to understand the properties of toric and generalized toric codes over ${\mathbb F}_q$. The minimum distance of these codes is strongly tied to the collections of lines in the finite geometry that contain subsets of the exponent vectors of the monomials that are evaluated to produce the standard generator matrix for the code. We argue that this connection is, in fact, even more direct than the connection with the lattice geometry of those exponent vectors considered as elements of ${\mathbb Z}^2$ or ${\mathbb R}^2$. This point of view should be useful both as a way to visualize properties of these codes and as a guide to heuristic searches for good codes constructed in this fashion. In particular, we will use these ideas to see a reason why these constructions have been so successful over the field ${\mathbb F}_8$, but less successful in other cases. This corrected version (posted on 3/6/2017) adds a necessary hypothesis in Proposition 2.4. A counterexample to the previous version was pointed out to the author by Melda G\"or\"ur and the author thanks her for bringing that to his attention. Theorem 3.2 and Corollary 3.3 are also reformulated so they do not refer to the new more restricted Proposition 2.4.