Existence results for doubly near resolvable (υ, 3, 2)-BIBDs |
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Authors: | E R LamkenS A Vanstone |
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Institution: | Department of Mathematics, Princeton University Fine Hall, Washington Rd, Princeton, NJ 08544-1000, USA Department of Combinatorics and Optimization, University of Waterloo, Waterloo, N2L 3G1 Ont., Canada |
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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={x1,x2,…,xυ}. (1) Each cell is either empty or contains a 3-subset of V. (2) Cell (xi, xi) is empty for i=1, 2,…, υ. (3) Row xi of F contains each element of V?{xi} once and column xi of F contains each element of V?{xi} 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. |
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