Perfect difference families, perfect difference matrices, and related combinatorial structures

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. They are not only theoretically important in their own rights, but also have many applications to other problems such as optical orthogonal codes, difference triangle sets, radar arrays, convolutional codes, and graceful labeling of graphs. In this talk, we mainly discuss 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 (mod 12), and that of an additive sequence of permutations of length 3 and order m, or briefly ASP(3,m), is m ≡ 1 (mod 2). So far, (12t + 1, 4, 1)-PDFs with t < 50 are known only for t = 1, 4− 33, 36, 41 with two definite exceptions of t = 2, 3, and ASP(3,m)’s with odd 3 < m < 200 are known only for m = 5, 7, 13, 15, 17, 19, 25, 35, 49, 65, 75, 85, 91, 95, 105, 119, 125, 133, 169, 175, 195 with two definite exceptions of m = 9, 11. The main idea of this talk is to use perfect difference families and additive sequences of permutations with “holes”. As an illustration of the power of the main idea, we show that a (12t+1, 4, 1)-PDF exists for any t ≤ 1000 except for t = 2, 3, and an ASP(3,m) exists for any odd 3 < m < 200 except for m = 9, 11 and possibly for m = 59. ∗This is a joint work with Ying Miao and Xianwei Sun.