New Methods in Coding Theory: Error-Correcting Codes and the Shannon Capacity

In this thesis we present several new results in coding theory, concerning error-correcting codes and the Shannon capacity.
We discuss a general symmetry reduction of matrices occurring in semidefinite programs in coding theory. We apply the symmetry reduction to efficiently compute semidefinite programming upper bounds for nonbinary error-correcting codes equipped with the Hamming distance, with the Lee distance and for binary constant weight codes. We also explore other methods to find new upper bounds for nonbinary codes with the Hamming distance, based on combinatorial divisibility arguments.
We study uniqueness and classification of codes, using the output of the semidefinite programming solver. Most of our classification results are related to subcodes of the binary Golay code.
Finally, we consider the Shannon capacity of circular graphs. The circular graph C_{d,q} is the graph with vertex set \Z_q (the cyclic group of order q) in which two distinct vertices are adjacent if and only if their distance (mod q) is strictly less than d. The Shannon capacity of C_{d,q} can be seen to only depend on the fraction q/d. We prove that the function which assigns to each rational number q/d the Shannon capacity of C_{d,q} is continuous at integer points q/d. Moreover, we give a new lower bound on the Shannon capacity of the 7-cycle.