可分集与几乎可分集的新构造  

New constructions for partitionable sets and almost partitionable sets

作　　者：周君灵[1] 常彦勋[1] Junling Zhou;Yanxun Chang

机构地区：[1]北京交通大学数学研究所,北京100044

出　　处：《中国科学：数学》2023年第2期407-418,共12页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11971053和12171028);北京市自然科学基金(批准号:1222013)资助项目。

摘　　要：可分集(partitionable set,PS)与几乎可分集(almost partitionable set,APS)是组合设计理论中两类重要的组合构型,与许多其他组合结构具有密切联系,如Z-循环Whist竞赛图、循环差阵、不含邻点的循环平衡样本设计、不交差族及光正交码等.由于可分集与几乎可分集的要求比较严苛,其存在性问题迄今远未解决.本文针对p≡7 (mod 8)为素数的情形,建立p2阶可分集与p阶几乎可分集的新构造方法,给出两类组合构型存在性的若干新结果.特别地,对于p≡7 (mod 8)的素数p,本文确定p<30,000的绝大部分p2阶PS的存在性,给出特定条件下p阶APS的存在性和渐近存在性,并得到p<50,000除去16个可能例外的p阶APS的存在性.Partitionable sets(PS) and almost partitionable sets(APS) are two important types of combinatorial configurations in the design theory.They have intimate connections with other combinatorial structures such as Zcyclic patterned starter Whist tournaments,cyclic difference matrices,cyclic balanced sampling plans excluding contiguous units,disjoint difference families,and optical orthogonal codes.The existing problems of PS and APS remain far from being settled as they demand stringent requirements.In this paper,we focus on the case where p≡7(mod 8) is a prime and establish new constructions for partitionable sets of order p~2 and almost partitionable sets of order p.In particular,for p≡7(mod 8),we show the existence of PS of order p~2 for a large portion of primes p<30,000,the existence and asymptotic existence of APS under certain conditions,and also the existence of APS of order p for all primes p <50,000 with 16 possible exceptions.

关 键 词：可分集 几乎可分集 Whist竞赛图 分圆类 

分 类 号：O157[理学—数学]