Self-dual codes and modules for finite groups in characteristic two
Self-dual codes and modules for finite groups in characteristic two
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DOI：
10.1109/tit.2004.831851
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发表时间：
2004-08
影响因子：
2.5
通讯作者：
C. Martínez‐Perez;W. Willems
中科院分区：
文献类型：
--
作者：
C. Martínez‐Perez;W. Willems
利用表示论的方法，我们研究了特征2中的自对偶群码及其扩张。我们证明了自对偶扩展群码的存在在很大程度上依赖于群代数KG的一种特殊结构，而这种特殊结构可以用初等数论中一个易于处理的准则来检验。令人惊讶的是，在二进制情况下，即使Gleason定理的逆成立，这样的码也是加倍的，即码的长度可以被8整除。此外，我们给出了Sloane和Thompson早期结果的一个简短的表示理论证明，该结果表明，即使G的Sylow 2-子群是循环的，二元自对偶码也永远不是加倍的。证明了完全在循环或Klein 4群为Sylow 2-子群的情况下，双重偶数群码是不存在的。
Using representation theoretical methods we investigate self-dual group codes and their extensions in characteristic 2. We prove that the existence of a self-dual extended group code heavily depends on a particular structure of the group algebra KG which can be checked by an easy-to-handle criteria in elementary number theory. Surprisingly, in the binary case such a code is doubly even if the converse of Gleason's theorem holds true, i.e., the length of the code is divisible by 8. Furthermore, we give a short representation theoretical proof of an earlier result of Sloane and Thompson which states that a binary self-dual group code is never doubly even if the Sylow 2-subgroups of G are cyclic. It turns out that exactly in the case of a cyclic or Klein four group as Sylow 2-subgroup doubly even group codes do not exist.