Neighbors, neighbor graphs and invariant rings in coding theory

TL;DR

This paper explores neighbor graphs and invariant rings in coding theory, extending previous results on binary self-dual codes and demonstrating finite generation of weight enumerator rings for certain code types.

Contribution

It extends neighbor graph analysis to Type III and IV codes and shows the finite generation of weight enumerator rings for these codes in arbitrary genus.

Findings

Extended neighbor graph results to Type III and IV codes

Proved finite generation of weight enumerator rings for specific code types

Identified minimal generators up to degree 24 and genus 3

Abstract

In the present paper, we discuss the class of Type III and Type IV codes from the perspectives of neighbors. Our investigation analogously extends the results originally presented by Dougherty [8] concerning the neighbor graph of binary self-dual codes. Moreover, as an application of neighbors in invariant theory, we show that the ring of the weight enumerators of Type II code $d_{n}^{+}$ and its neighbors in arbitrary genus is finitely generated. Finally, we obtain a minimal set of generators of this ring up to the space of degree 24 and genus 3.