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(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, 4−36, 41, 46 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, 13-29, 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 (12t +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 415 416 GE, MIAO, AND SUN 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. q 2010 Wiley Periodicals, Inc. J Combin Designs 18: 2010 Keywords: perfect difference family; perfect system of difference sets; perfect difference packing; additive sequence of permutations; perfect difference matrix; semi-perfect group divisible design ‡ Supporting Information may be found in the online version of this article.

Semi-cyclic holey group divisible designs with block size three

“…They used a different terminology called h-perfect g-regular cyclic packing. For more information on perfect difference families, the interested reader may refer to [4,21,34].…”

Construction of High-Rate Regular Quasi-Cyclic LDPC Codes Based on Cyclic Difference Families

“…Since there are no PDFs for k ≥ 6 and no sufficiently many PDFs for k = 5 from Theorem 2, we focus on the case of k = 3, 4. The construction of PDFs for k = 3 and 4 is provided in [15] and [17], respectively.…”

More results on optimal optical orthogonal codes with weight four

“…If = ( − 1) + 1, then blocks The following result is from [18]. Perfect difference families were also used to construct optimal OOCs, see [2], [3] for the examples.…”

“…Perfect difference families were also used to construct optimal OOCs, see [2], [3] for the examples. For more details on perfect difference families, we refer to [1], [18] and the references therein. Remark 1 It is not difficult to see that if ℬ is a ( , , 1)-PDF, then ℬ is also an optimal CP( , 1; + ) for each 1 ≤ < ( − 1).…”