**Journal of Combinatorial Designs**

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

## Abstract

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, (12*t*+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<*m*<200 are known only for *m*=5,7,13−29,35,45,49,65,75,85,91,95,105,115,119,121,125,133,135,145,147,161,169,175,189,195 with two definite exceptions of *m*=9,11. In this article, we show that a (12*t*+1,4,1)-PDF exists for any *t*⩽1,000 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. The main idea of this article is to use perfect difference families and additive sequences of permutations with “holes”. We first introduce the concepts of an incomplete perfect difference matrix with a regular hole and a perfect difference packing with a regular difference leave, respectively. We show that an additive sequence of permutations is in fact equivalent to a perfect difference matrix, then describe an important recursive construction for perfect difference matrices via perfect difference packings with a regular difference leave. Plenty of perfect difference packings with a desirable difference leave are constructed directly. We also provide a general recursive construction for perfect difference packings, and as its applications, we obtain extensive recursive constructions for perfect difference families, some via incomplete perfect difference matrices with a regular hole. Examples of perfect difference packings directly constructed are used as ingredients in these recursive constructions to produce vast numbers of perfect difference families with block size 4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 415–449, 2010