Laplacian Simplices II: A Coding Theoretic Approach

TL;DR

This paper explores the relationship between Laplacian simplices derived from graphs and associated linear codes, advancing the understanding of their properties and constructing new classes of error-correcting codes.

Contribution

It introduces a duality-preserving code construction from reflexive Laplacian simplices and investigates their properties for specific graph classes, including cycles and complete graphs.

Findings

Constructed a duality-preserving linear code from reflexive Laplacian simplices.

Provided a systematic analysis of codes for cycles, complete graphs, and prime-vertex graphs.

Developed an asymptotically good family of MDS codes with arbitrary rational rates.

Abstract

This paper further investigates \emph{Laplacian simplices}. A construction by Braun and the first author associates to a simple connected graph $G$ a simplex $\cP_G$ whose vertices are the rows of the Laplacian matrix of $G$. In this paper we associate to a reflexive $\cP_G$ a duality-preserving linear code $\cC(\cP_G)$. This new perspective allows us to build upon previous results relating graphical properties of $G$ to properties of the polytope $\cP_G$. In particular, we make progress towards a graphical characterization of reflexive $\cP_G$ using techniques from Ehrhart theory. We provide a systematic investigation of $\cC(\cP_G)$ for cycles, complete graphs, and graphs with a prime number of vertices. We construct an asymptotically good family of MDS codes. In addition, we show that any rational rate is achievable by such construction.