The Existence of Doubly Disjoint \((mt + 1, m, m – 1)\) Difference Families

Abstract

A \((v,m,m-1)\)-BIBD \(D\) is said to be near resolvable (NR-BIBD) if the blocks of \(D\) can be partitioned into classes \(R_1, R_2, \ldots, R_v\) such that for each point \(x\) of \(D\), there is precisely one class having no block containing \(x\) and each class contains precisely \(v – 1\) points of the design. If a \((v,m,m-1)\)-NRBIBD has a pair of orthogonal near resolutions, it is said to be doubly resolvable and is denoted DNR\((v,m,m-1)\)-BIBD. A lot of work had been done for the existence of \((v,m,m-1)\)-NRBIBDs, while not so much is known for the existence of DNR\((v,m,m-1)\)-BIBDs except for the existence of DNR\((v,3,2)\)-BIBDs. In this paper, doubly disjoint \((mt+1,m,m-1)\) difference families \(((mt+1,m,m-1)\)-DDDF in short) which were called starters and adders in the previous paper by Vanstone, are used to construct DNR\((v,m,m-1)\)-BIBDs. By using Weil’s theorem on character sum estimates, an explicit lower bound for the existence of a \((mt+1,m,m-1)\)-DDDF and a DNR\((mt+1,m,m-1)\)-BIBD is obtained, where \(mt+1\) is a prime power, \((m,t)=1\). By using this result, it is also proved that there exist a \((v,4,3)\)-DDDF and a DNR\((v,4,3)\)-BIBD for any prime power \(v\equiv 5\pmod{8}\) and \(v\geq 5d\).