Semidefinite Programming, Binary Codes and a Graph Coloring Problem

Experts in information theory have long been interested in the maximal size A(n,d) of a binary error-correcting code of length n and distance d, The problem of determining A(n,d) involves both the construction of good codes and the search for good upper bounds. From 1973 to 2003, the linear programming bound found many applications, but there were few significant theoretical advances until Schrijver proposed a new code upper bound via semidefinite programming. Using the Terwilliger algebra, Schrijver formulated a new SDP strengthening of the LP approach. In this book we study the dual solutions of the semidefinite program bound, and explore the combinatorial meaning of these variables for small n and d. To obtain information like this, we wrote a program with both MATLAB and CVX modules to get solution of our primal SDP formulation. Our program efficiently generates the primal solutions with corresponding constraints for any n and d. We also wrote programs to parse the optimal solutions of the primal SDP and obtain the optimal solutions of the dual problem automatically. These values are very useful for later study of the combinatorial meaning of such solutions.