  In this work, we introduce 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 gamma values. This completes the theoretical
list of possible values for gamma in W_{68,2}.
