Regularly Cyclic Packing Of Block Size 5

A cycilc packing with order v, block size k, and indexλ= 1, or a CP(k,1;v), can be defined equivalently as a family B = {B₁, B₂,…, B_t} of t k－subsets (base blocks) of Z_v, where B_i = {b_i1, b_i2,..., b_ik}, 1≤i≤t, such that the difference in B,ΔB = {b_ij－b_is : 1≤i≤t, j≠s, 1≤j, s≤k}, cover each non-zero residue of integers modulo v at most once. The set Z_v\Δ(B) is defined the difference leave of the CP(k,1;v), denoted by DL(B). If DL(B) is a subgroup of Z_v having order g, B is termed g-regular cycilc packing.If 1≤g≤k(k－1), a g-regular CP(k,1;v) is called optimal. An optimal CP(k,1;v) is a set-theoretic characterization of an optimal optical orthogonal codes of length v, weight k and auto/cross-correlation unity. The existence of g-regular CP(k,1;v)'s has attracted considerable attention in design theory. This is because they not only have mathematical interest but also have nice application in digital communication. (see[1, 2, 3, 4, 5, 6, 7, 8, 9]).In this paper, we obtin the following two results by mainly use direct and recusive constructions.Theorem 2.9 For any prime p congruent to 1 modulo 6, 20-regular CP(5,1; 20p) exists. Theorem 3.5 Let u is a product of primes p congruent to 1 modulo 6, for any non-negative integerα, (20·3^αu, 5,1)－OOC exists.