Existence results for doubly near resolvable (upsilon, 3, 2)-BIBDs

Abstract Let V be a set of υ elements. A (1, 2; 3, υ, 1)-frame F is a square array of side v which satisfies the following properties. We index the rows and columns of F with the elements of V , V ={ x 1 , x 2 ,…, x υ }. (1) Each cell is either empty or contains a 3-subset of V . (2) Cell ( x i , x i ) is empty for i =1, 2,…, υ . (3) Row x i of F contains each element of V −{ x i } once and column x i of F contains each element of V −{ x i } once. (4) The collection of blocks obtained from the nonempty cells of F is a (υ, 3, 2)-BIBD. A (1, 2; 3, υ, 1)-frame is a doubly near resolvable (υ, 3, 2)-BIBD. In this paper, we first present a survey of existence results on doubly near resolvable (υ, 3, 2)-BIBDs and (1, 2; 3, υ, 1)-frames. We then use frame constructions to provide a new infinite class of doubly near resolvable (υ, 3, 2)-BIBDs by constructing (1, 2; 3, υ, 1)-frames.