Minimum distance of linear codes and the $\alpha$-invariant

TL;DR

This paper explores the relationship between the minimum distance of linear codes and the $eta$-invariant of associated modules using commutative algebra techniques.

Contribution

It introduces a novel algebraic approach linking code minimum distance with the $eta$-invariant via graded modules and homological algebra.

Findings

Established a connection between minimum distance and the $eta$-invariant.

Extended previous interpretations through algebraic methods.

Provided new tools for analyzing linear codes using algebraic invariants.

Abstract

The simple interpretation of the minimum distance of a linear code obtained by De Boer and Pellikaan, and later refined by the second author, is further developed through the study of various finitely generated graded modules. We use the methods of commutative/homological algebra to find connections between the minimum distance and the $\alpha$-invariant of such modules.