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Some codes in symmetric and linear groups
Discrete Mathematics ( IF 0.7 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.disc.2019.111719
Holly M. Green , Martin W. Liebeck

Abstract For a finite group G , a positive integer λ , and subsets X , Y of G , write λ G = X Y if the products x y ( x ∈ X , y ∈ Y ), cover G precisely λ times. Such a subset Y is called a code with respect to X , and when λ = 1 it is a perfect code in the Cayley graph Cay ( G , X ). In this paper we present various families of examples of such codes, with X closed under conjugation and Y a subgroup, in symmetric groups, and also in special linear groups S L 2 ( q ) . We also propose conjectures about the existence of some much wider families.

中文翻译:

对称群和线性群中的一些代码

摘要 对于有限群 G 、正整数 λ 和 G 的子集 X , Y ,如果乘积 xy ( x ∈ X , y ∈ Y ),精确覆盖 G λ 次,记作 λ G = XY。这样的子集 Y 称为关于 X 的代码,当 λ = 1 时,它是 Cayley 图 Cay ( G , X ) 中的完美代码。在本文中,我们展示了此类代码的各种示例族,其中 X 在共轭下闭合,Y 是一个子群,在对称群中,也在特殊线性群 SL 2 ( q ) 中。我们还提出了关于一些更广泛的家庭的存在的猜想。
更新日期:2020-08-01
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