On self-dual, doubly even codes of length 32
On self-dual, doubly even codes of length 32
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长度为 32 的自对偶、双偶码
DOI：
10.1016/0097-3165(89)90077-0
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发表时间：
1989
期刊：
影响因子：
--
通讯作者：
H. Koch
中科院分区：
文献类型：
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作者：
H. Koch
在一篇联合论文[2]中，Conway和Pless研究了长度为32的二进制线性自对偶双偶码，并表明存在85个不相等的这种码（双偶意味着所有码字的权重都是4的倍数）。毫无疑问，其中最有趣的是最小重量的活代码。这些是Sloane[4]术语中长度为32的II型极值码。下面我们称之为cp代码。[Z]中的证明方法包括通过几个过程（包括占卜）找到密码，然后通过计数公式表明已经发现了所有的密码。事实上，这种方法是非常费力的，因为必须计算码的自同构群，如果只对极值码感兴趣，则必须花很长时间来研究所有其他80个码。实际上，[2]只给出了证明方法的描述。在本文中，我们用其他的证明方法给出了一个完整的和相对简短的证明，证明了不平衡的cp码恰好有5个，这也进一步说明了这些码的结构。我们从对活cp代码的描述开始。其中两个是在Conway和Pless研究之前就已知的，即p= 31的扩展二次剩余码和van Lint[3]符号中的Reed-Muller码W（2,5）。我们在这里分别用QR和RM表示这些代码。
In a joint paper [2] Conway and Pless studied binary linear self-dual doubly even codes of length 32 and showed that there are 85 inequivalent such codes (doubly even means that the weights of all code words are multiples of 4). Without doubt the most interesting of them are the live codes of minimal weight 8. These are the extremal codes of type II and length 32 in the terminology of Sloane [4]. We call them CP-codes in the following. The method of proof in [Z] consists in finding codes by several processes “including divination” and then to show by means of the counting formula that one has discovered all the codes. In fact this method is very laborious since one has to compute the automorphism groups of the codes, and if one is only interested in the extremal codes, one has nevertheless to go the long way through all the other 80 codes. In fact,[2] gives only a description of the method of the proof. In the present paper we give a full and relatively short proof that there are exactly five inequilivalent CP-codes using other methods of proof, which also show something more about the architecture of these codes. We begin with a description of the live CP-codes. Two of them were known before the investigations of Conway and Pless, namely the extended quadratic residue code for p= 31 and the Reed-Muller code W (2, 5) in the notation of van Lint [3]. We denote these codes here by QR and RM, respectively.