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 $W_{68,2}$.

中文翻译：

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来自广义邻居的长度为68的新的极值二进制自对偶代码

在这项工作中，我们使用了自我对偶代码之间的距离的概念，它概括了自我对偶代码的邻居的概念。使用k邻居，我们能够使用新的权重枚举器构造长度为68的极值二进制自对偶代码。我们使用新的权重枚举器构造143个长度为68的极值二进制自对偶代码，其中包括42个代码，$\gamma =8$ 在他们 ${\mathrm{w\; ^}}_{68，2}$ 和40与 $\gamma =9$ 在他们 ${\mathrm{w\; ^}}_{68，2}$。这些示例是有关这些γ值的文献中的第一个。这样就完成了可能值的理论名单γ在${\mathrm{w\; ^}}_{68，2}$。