A projective two-weight code related to the simple group ${\rm Co}_1$ of Conway
A projective two-weight code related to the simple group ${\rm Co}_1$ of Conway
复制标题
与 Conway 的简单群 ${ m Co}_1$ 相关的投影二重码
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
B. Rodrigues
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文献类型：
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作者：
B. Rodrigues
本文构造了与Conway的偶发单群${\rm Co}_1$有关的二元$[98280,24,47104]_2$投影二权码，它是由${\rm Co}_1$对${\rm Co}_2$的集的原元作用所引起的置换模的忠实且绝对不可约的子模。该码的对偶码是一个均匀包装的$[98280,98256,3]_2$码。码字的几何意义可以追溯到Leech格中的向量，从而揭示了码中任意非零权码字的稳定器是${\rm Co}_1$的极大子群。同样，对偶码中权值最小码字的稳定器是${\rm Co}_1$的一个极大子群。利用编码的码字，构造了一个顶点为16777216、价码为98280的强正则图。
A binary $[98280, 24, 47104]_2$ projective two-weight code related to the sporadic simple group ${\rm Co}_1$ of Conway is constructed as a faithful and absolutely irreducible submodule of the permutation module induced by the primitive action of ${\rm Co}_1$ on the cosets of ${\rm Co}_2$. The dual code of this code is a uniformly packed $[98280, 98256,3]_2$ code. The geometric significance of the codewords of the code can be traced to the vectors in the Leech lattice, thus revealing that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of ${\rm Co}_1$. Similarly, the stabilizer of the codewords of minimum weight in the dual code is a maximal subgroup of ${\rm Co}_1$. As by-product, a new strongly regular graph on 16777216 vertices and valency 98280 is constructed using the codewords of the code.