A Graph-Theoretic Approach to Bounds for Error-Correcting Codes CIMPA-UNESCO-PHILIPPINES Summer School on Semidefinite Programming in Algebraic Combinatorics

In these notes, we address bounds for error-correcting codes. Our approach is from the viewpoint of algebraic graph theory. We therefore begin with a review of the algebraic structure of the Hamming graph, focusing on the binary case. We then derive Delsarte’s linear programming bound and explore some applications of it. In the second part of the notes, we introduce Terwilliger’s subconstituent algebra and explore its connection to the semidefinite programming approach of Schrijver. Throughout, our focus is on the binary case. The three steps presented here are a summary of the structure of the Terwilliger algebra as presented by Go, a surprising connection to the biweight enumerator of a binary code, and a full characterization of the positive semidefinite cone of the algebra, given by Visentin and Martin.