Perfect cyclic designs

AbstractLet k and λ be positive integers and let v be a positive integer or an infinite cardinal number. A cyclic design in the class D(v, k, λ) consists of a set Q of v elements and a collection of cyclically ordered k -subsets of Q called cyclic blocks such that every ordered pair of elements of Q are consecutive in exactly λ cyclic blocks. (Note the block {a1, a2, a3, a4,…, ak} has the cyclic order a1 < a2 < a3 < a4 … < a8 < al and a1ai+l are consecutive. Also a 1ai+l are said to be t apart in the block.) If in addition for i = 1, 2 ,…, k − 1 every ordered pair of elements are i apart in exactly λ of the blocks we say the design is perfect and belongs to the class PD(v, k, λ). The following theorems are proved. Let q1, q2,…, ql be distinct prime powers such that each of the corresponding Galois fields contains a primitive kth root of unity, k ≠ 1, 2. Then there exists a design in the class PD(v, k, 1) where v = q1q2⋯ql. If, furthermore, there is a pairwise balanced block design with λ = 1 and block sizes amongst the numbers q1, q2, …, ql on a set of w elements then there is a design in the class PD(w, k, 1). Also if K is an algebraically closed field of cardinality v (infinite) then there are designs in the class PD(v, k, 1) for every integer k. Finally, if k is an odd prime p, then PD(v, p, 1) exists for all v such that v