Doubly near resolvable m-cycle systems

An m-cycle system C of λKn is said to be near resolvable if the mcycles in C can be partitioned into near parallel classes R1, R2, . . . , Rλn/2, each one of which is a 2-factor of λKn − v for some vertex v in λKn. If an NR(n,m, λ)-CS has a pair of orthogonal resolutions, it is said to be doubly resolvable and is denoted by DNR(n,m, λ)-CS. For m = 2, 3, DNR(n,m, 2)-CSs are known as Room frames of type 1 and DNR(n, 3, 2)-BIBDs, respectively. The spectra for Room frames of type 1 and DNR(n, 3, 2)-BIBDs have been completed. To our knowledge, very little is known about the existence of DNR(n,m, λ)-CSs with m ≥ 4. In this paper, we use Weil’s theorem on character sum estimates to give an implicit lower bound about the existence of DNR(mt + 1,m, λ)-CSs, where mt+1 is a prime power. From this result, we also show that there exists a DNR(n, 4, λ)-CS for any prime power n ≡ 1 (mod 4) and n ≥ 13.