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Self-dual codes with an automorphism of order 7 and s-extremal codes of length 68

机译：自对偶码为7阶自构，长度为s的极值码

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This paper studies and classifies optimal binary self-dual codes having an automorphism of order 7 with 9 cycles. This classification is done by applying a method for constructing binary self-dual codes with an automorphism of odd prime order p. There are exactly 69781 inequivalent binary self-dual [64, 32, 12] codes with an automorphism of type 7 - (9, 1). As for binary [66, 33, 12] self-dual codes with an automorphism of type 7 - (9, 3) there are 1652432 such codes. We also construct more than 4 million new optimal codes of length 68 among which are the first known examples of the very elusive s-extremal self-dual codes. We prove the nonexistence of [70, 35,14] codes with an automorphism of type 7 - (9, 7). Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before. (C) 2017 Published by Elsevier Inc.
机译：本文研究并分类了具有9个循环的7阶自同构性的最佳二进制自对偶码。通过应用一种构造具有奇质数阶p自同构的二进制自对偶码的方法来完成此分类。正好有69781个等价的二进制自对偶[64、32、12]码，其自同构类型为7-（9，1）。对于具有7-（9，3）类型自同构的二进制[66，33，12]自对偶代码，有1652432这样的代码。我们还构造了超过400万个长度为68的新最优代码，其中最难捉摸的s-极值自对偶代码是第一个已知示例。我们用类型7-（9，7）的自同构证明[70，35,14]代码不存在。所有长度的大多数构造代码都有权重枚举器，以前不知道其存在。 （C）2017由Elsevier Inc.发布

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