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Abelian difference sets with the symmetric difference property
Designs, Codes and Cryptography ( IF 1.6 ) Pub Date : 2021-01-16 , DOI: 10.1007/s10623-020-00829-5
James A. Davis , J. J. Hoo , Connor Kissane , Ziming Liu , Calvin Reedy , Kartikey Sharma , Ken Smith , Yiwei Sun

A $$(v,k,\lambda )$$ ( v , k , λ ) symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. The designs associated to the symplectic difference sets introduced by Kantor (J Algebra 33:43–58, 1975) have the SDP. Parker (J Comb Theory Ser A 67:23–43, 1994) claimed that the symplectic design on 64 points is the only SDP design on 64 points admitting an abelian regular automorphism group (an abelian difference set). We show in this paper that there is an SDP design on 64 points that is not isomorphic to the symplectic design and yet admits the group $$C_8 \times C_4 \times C_2$$ C 8 × C 4 × C 2 as a regular automorphism group. This abelian difference set is the first in an infinite family of abelian difference sets whose designs have the SDP and yet are not isomorphic to the symplectic designs of the same order. We define a new method for establishing the non-isomorphism of the two families.

中文翻译:

具有对称差分性质的阿贝尔差分集

$$(v,k,\lambda )$$ ( v , k , λ ) 对称设计如果任何三个块的对称差异是一个块或 a 的补集,则称其具有对称差异特性 (SDP)堵塞。Kantor (J Algebra 33:43–58, 1975) 引入的与辛差分集相关的设计具有 SDP。Parker (J Comb Theory Ser A 67:23–43, 1994) 声称 64 点上的辛设计是唯一承认阿贝尔正则自同构群(阿贝尔差分集)的 64 点上的 SDP 设计。我们在本文中展示了 64 点上的 SDP 设计,它与辛设计不同构,但承认群 $$C_8 \times C_4 \times C_2$$ C 8 × C 4 × C 2 作为正则自同构团体。这个阿贝尔差分集是无限阿贝尔差分集家族中的第一个,其设计具有 SDP,但与同阶辛设计不同构。我们定义了一种新的方法来建立两个家族的非同构。
更新日期:2021-01-16
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