On coset leader graphs of structured linear codes

TL;DR

This paper introduces a new elementary approach using coset leader graphs with high discrete Ricci curvature to derive bounds on locally correctable and testable binary linear codes, improving some existing bounds.

Contribution

It presents a novel method leveraging coset leader graphs and Ricci curvature to analyze code properties, offering improved bounds for certain locally testable codes.

Findings

Bounds for locally correctable codes are better than previous information-theoretic bounds.

Bounds for a family of locally testable codes are improved.

The approach is elementary compared to quantum information methods.

Abstract

We suggest a new approach to obtain bounds on locally correctable and some locally testable binary linear codes, by arguing that these codes (or their subcodes) have coset leader graphs with high discrete Ricci curvature. The bounds we obtain for locally correctable codes are worse than the best known bounds obtained using quantum information theory, but are better than those obtained using other methods, such as the "usual" information theory. (We remark that our methods are completely elementary.) The bounds we obtain for a family of locally testable codes improve the best known bounds.