New extremal binary self-dual codes of length 68 from generalized neighbors

In this work, we use the concept of distance between self-dual codes, which generalizes the concept of a neighbor for self-dual codes. Using the k-neighbors, we are able to construct extremal binary self-dual codes of length 68 with new weight enumerators. We construct 143 extremal binary self-dual codes of length 68 with new weight enumerators including 42 codes with $\gamma =8$ in their $W_{68,2}$ and 40 with $\gamma =9$ in their $W_{68,2}$. These examples are the first in the literature for these γ values. This completes the theoretical list of possible values for γ in $W_{68,2}$.