Enumeration of the Binary Self-Dual Codes of Length 34

An \( (n,k) \) binary self-orthogonal code is an \( (n,k) \) binary linear code \( C \) that is contained in its orthogonal complement \( C^\bot \). A self-orthogonal code \( C \) is self-dual if \( C = C^\bot \). Two codes, \( C_1 \) and \( C_2 \), are \emph{equivalent} if and only if there exists a coordinate permutation of \( C_1 \) that takes \( C_1 \) into \( C_2 \). The automorphism group of a code \( C \) is the set of all coordinate permutations of \( C \) that takes \( C \) into itself.

This paper is a continuation of the work presented in [2], in which we described an algorithm for enumerating inequivalent binary self-dual codes. We used our algorithm to enumerate the self-dual codes of length up to and including 32. Our algorithm also found the size of the automorphism group of each code.

We have since made several improvements to our algorithm. It now generally runs faster. It also now finds generators for the automorphism group of each code. We have used our improved algorithm to enumerate the self-dual codes of length 34. We have also found the automorphism groups for each of our self-dual codes of length less than or equal to 34. The list of length 34 codes are new, as are the lists of automorphism groups for the length 32 and length 34 codes. We have found there are 19914 inequivalent length 34 codes with distance 4 and 938 length 34 codes with distance 6.