On incomplete and balanced incomplete block designs

Abstract

By an incomplete block design we will mean an arrangement of v different varieties of objects in b distinct blocks or sets such that a block does not contain all the varieties and a variety appears at most once in a block. An incomplete block design which satisfies two extra conditions, viz. (i) every pair of varieties occurs together in ' S 0 of the blocks and (ii) each block contains the same number of objects, say k, is called a balanced incomplete block (b.i.b. for conciseness) design. A b.i.b. design with b=v is sometimes called a v-k-X configuration. It is known [1] and can be seen below that in a b.i.b. design every variety occurs the same number, say r, of times. Clearly then bk = vr, X(v-1) =r(k-1). For a b.i.b. design Fisher's inequality b_v must also hold [1 ]. An incomplete block design satisfying condition (i) is not necessarily a b.i.b. design. Ryser [3 ] has proved (in an essentially equivalent form) that if in a symmetrical incomplete block design (that is to say, one in which b = v) every variety appears a constant number of times, say r, also condition (i) holds, then X(v 1) = r(k 1) and (ii) holds and it is a symmetrical b.i.b. design. We give here conditions under which any incomplete block design becomes a b.i.b. design. An extension of Ryser's result in another direction has been given in [2].