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In this thesis we present several results in coding theory, concerningerror-correcting codes and the Shannon capacity.  1. We give a general symmetry reduction of matrices occuring in semidefiniteprograms in coding theory.  2. We apply the symmetry reduction to efficiently compute semidefiniteprogramming upper bounds for nonbinary error-correcting codes equipped with theHamming distance (joint work with Bart Litjens and Lex Schrijver), with the Leedistance and for binary constant weight codes.  3. We explore other methods to find new upper bounds for nonbinary codes withthe Hamming distance, based on combinatorial divisibility arguments.  4. We study uniqueness and classification of codes, using the output of thesemidefinite programming solver. Most of our classification results are relatedto subcodes of the binary Golay code. This is joint work with Andries Brouwer.  5. We consider the Shannon capacity of circular graphs. The circular graph$C_{d,q}$ is the graph with vertex set $\mathbb{Z}_q$ (the cyclic group oforder q) in which two distinct vertices are adjacent if and only if theirdistance (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 functionwhich 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 onthe Shannon capacity of the $7$-cycle. This is joint work with Lex Schrijver.  This thesis was written at the University of Amsterdam under supervision ofLex Schrijver.

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