Denniston 偏差集在奇素数情形下存在
组合数学
2024-10-08 v3
摘要
Denniston 在阶为 2^{3m} 的初等阿贝尔群中，对所有 m ≥ 2, 1 ≤ r < m，构造了参数为 (2^{3m}, (2^{m+r} - 2^m + 2^r)(2^m-1), 2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2), (2^{m+r}-2^m+2^r)(2^r-1)) 的偏差集（PDSs）。这些对应于偶阶 Desargues 射影平面中的极大弧。本文中，我们表明——尽管奇阶 Desargues 射影平面中不存在极大弧——参数为 (p^{3m}, (p^{m+r} - p^m + p^r)(p^m-1), p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2), (p^{m+r}-p^m+p^r)(p^r-1)) 的 PDSs 在所有阶为 p^{3m} 的初等阿贝尔群中存在，其中 m ≥ 2, r ∈ {1, m-1}，p 为奇素数，并给出一种构造。我们的方法使用由分圆类之并构成的 PDSs。
引用
@article{arxiv.2311.00512,
title = {Denniston partial difference sets exist in the odd prime case},
author = {James A. Davis and Sophie Huczynska and Laura Johnson and John Polhill},
journal= {arXiv preprint arXiv:2311.00512},
year = {2024}
}
备注
Since our work was announced, we have become aware that an equivalent result has simultaneously been proved by de Winter for projective two-weight sets, and a corresponding coding theory result was proved by Bierbrauer and Edel in 1997; references and citations have been added for these