Small polygons and toric codes

TL;DR

This paper presents two methods for classifying plane lattice polygons and analyzes the toric codes they produce, achieving record-breaking minimum distances over small finite fields, including a notable code over F_7.

Contribution

It introduces systematic classification approaches for plane lattice polygons and demonstrates their effectiveness in generating high-performance toric codes with improved minimum distances.

Findings

Achieved a [36,19,12]-code over F_7 with record minimum distance

Matched or exceeded the best known minimum distances for toric codes over small fields

Provided two different classification methods for plane lattice polygons

Abstract

We describe two different approaches to making systematic classifications of plane lattice polygons, and recover the toric codes they generate, over small fields, where these match or exceed the best known minimum distance. This includes a [36,19,12]-code over F_7 whose minimum distance 12 exceeds that of all previously known codes.