AG Codes from Polyhedral Divisors

TL;DR

This paper extends the theory of toric codes to a broader class called T-varieties, providing formulas for divisors and global sections, and demonstrating improved code constructions with potential for better error-correcting capabilities.

Contribution

It introduces a new framework for constructing evaluation codes from T-varieties, generalizing toric and AG codes, with explicit formulas and improved code examples.

Findings

Codes on ruled surfaces outperform related product codes

Intersection theory estimates improve minimum distance bounds

More sophisticated T-varieties lead to better codes

Abstract

A description of complete normal varieties with lower dimensional torus action has been given by Altmann, Hausen, and Suess, generalizing the theory of toric varieties. Considering the case where the acting torus T has codimension one, we describe T-invariant Weil and Cartier divisors and provide formulae for calculating global sections, intersection numbers, and Euler characteristics. As an application, we use divisors on these so-called T-varieties to define new evaluation codes called T-codes. We find estimates on their minimum distance using intersection theory. This generalizes the theory of toric codes and combines it with AG codes on curves. As the simplest application of our general techniques we look at codes on ruled surfaces coming from decomposable vector bundles. Already this construction gives codes that are better than the related product code. Further examples show that we can improve these codes by constructing more sophisticated T-varieties. These results suggest to look further for good codes on T-varieties.