Laplacian Simplices II: A Coding Theoretic Approach
Abstract
This paper further investigates Laplacian simplices. A construction by Braun and the first author associates to a simple connected graph G a simplex G whose vertices are the rows of the Laplacian matrix of G. In this paper we associate to a reflexive G a duality-preserving linear code (G). This new perspective allows us to build upon previous results relating graphical properties of G to properties of the polytope G. In particular, we make progress towards a graphical characterization of reflexive G using techniques from Ehrhart theory. We provide a systematic investigation of (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.
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