Perfect difference families, perfect difference matrices, and related combinatorial structures
Perfect difference families, perfect difference matrices, and related combinatorial structures
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DOI：
10.1002/jcd.20259
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发表时间：
2010-11
影响因子：
0.7
通讯作者：
G. Ge;Y. Miao;Xianwei Sun
中科院分区：
文献类型：
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作者：
G. Ge;Y. Miao;Xianwei Sun
区组大小为k，k=4，5的完全差族的存在性问题和长度为n，n=3，4的置换的加性序列的存在性问题是组合设计理论30多年来的两个突出的公开问题。本文主要研究了块大小为k=4的完美差族和长度为n=3的置换的加性序列。存在大小为4的块大小为v的完全差族，或简写为(v，4，1)-pdf的必要条件是v≡1(Mod12)，而长度为3的m阶置换的加法序列，或简略为asp(3，m)，存在的必要条件是m≡1(Mod2)。到目前为止，带有t<；50的(12t+1，4，1)-−仅对t=1，4 PDF 36，41，46已知，但有两个明确的例外：t=2，3和具有奇3的ASP(3，m)‘S
The existence problems of perfect difference families with block size k, k=4,5, and additive sequences of permutations of length n, n=3,4, are two outstanding open problems in combinatorial design theory for more than 30 years. In this article, we mainly investigate perfect difference families with block size k=4 and additive sequences of permutations of length n=3. The necessary condition for the existence of a perfect difference family with block size 4 and order v, or briefly (v, 4,1)‐PDF, is v≡1(mod12), and that of an additive sequence of permutations of length 3 and order m, or briefly ASP (3, m), is m≡1(mod2). So far, (12t+1,4,1)‐PDFs with t<50 are known only for t=1,4−36,41,46 with two definiteexceptions of t=2,3, and ASP (3, m)'s with odd 3