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Partial difference sets in C2n×C2n
Discrete Mathematics ( IF 0.7 ) Pub Date : 2020-04-01 , DOI: 10.1016/j.disc.2019.111744
Martin E. Malandro , Ken W. Smith

Abstract Let G n denote the group C 2 n × C 2 n , where C k is the cyclic group of order k . We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in G n . We use our algorithm to obtain all of these PDS in G n for 2 ≤ n ≤ 9 , and we obtain partial results for n = 10 and n = 11 . Most of these PDS are new. For n ≤ 4 we also identify group-inequivalent PDS. Our approach involves constructing tree diagrams and canonical colorings of these diagrams. Both the total number and the number of group-inequivalent PDS in G n appear to grow super-exponentially in n . For n = 9 , a typical canonical coloring represents in excess of 1 0 146 group-inequivalent PDS, and there are precisely 2 520 reversible Hadamard difference sets.

中文翻译:

C2n×C2n 中的偏差分集

摘要 令 G n 表示群 C 2 n × C 2 n ,其中 C k 是 k 阶循环群。我们给出了一个算法来枚举 G n 中的规则非平凡偏差分集(PDS)。我们使用我们的算法在 2 ≤ n ≤ 9 的情况下获得 G n 中的所有这些 PDS,并且我们获得了 n = 10 和 n = 11 的部分结果。大多数这些 PDS 都是新的。对于 n ≤ 4,我们还确定了组不等价的 PDS。我们的方法涉及构建树图和这些图的规范着色。G n 中的总数和组不等价 PDS 的数量似乎在 n 中呈超指数增长。对于 n = 9 ,典型的规范着色表示超过 1 0 146 个组不等价 PDS,并且恰好有 2 520 个可逆的哈达玛差分集。
更新日期:2020-04-01
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