Remarks on generalized toric codes

TL;DR

This paper investigates how the minimum distance of generalized toric codes depends on the base field and demonstrates that, in some cases, it matches that of the convex hull of the defining set, with an example of a superior code over GF(8).

Contribution

It provides new insights into the relationship between the base field size and the minimum distance of generalized toric codes, including a specific example surpassing known codes.

Findings

Minimum distance can match that of the convex hull in large fields.

A [49,12,28] code over GF(8) outperforms previous codes.

The results link geometric properties of the defining set to code parameters.

Abstract

This note presents some new information on how the minimum distance of the generalized toric code corresponding to a fixed set of integer lattice points S in R^2 varies with the base field. The main results show that in some cases, over sufficiently large fields, the minimum distance of the code corresponding to a set S will be the same as that of the code corresponding to the convex hull of S. In an example, we will also discuss a [49,12,28] generalized toric code over GF(8), better than any previously known code according to M. Grassl's online tables, as of July 2011.