described a sufficient condition for the non-embeddability of a quasi-derived design into a symmetric balanced incomplete block design. In this paper, by using the notion of incomplete designs, this criterion is changed to find certain quasi-derived designs in some special cases. Many infinite series of non-embeddable quasi-derived designs are thus constructed.
1996 Academic Press, Inc.
1. Introduction
We assume that the reader is familiar with some basic concepts in design theory. For general information and notations see [1].
Given a symmetric balanced incomplete block (BIB) design D, SB(k, *; v), and a block B in D, removing B and all its treatments from the remaining blocks yields a BIB design B(k&*, *; v&k) which is called residual (with respect to B). Similarly, the treatments of B and the intersections of B with the remaining blocks form a BIB design B(*, *&1; k) which is called derived. In general a BIB design B(k, *; v) with parameters v, b, r, k, * satisfying r=k+* is called quasi-residual, while a BIB design B(k, *; v) with *=k&1 is called quasi-derived.
Van Lint [2] made a systematic investigation about the problem of the non-trivial non-embeddability of the quasi-residual and quasi-derived designs, i.e., given a quasi-residual or quasi-derived design, whether it is embeddable as a residual or derived design into a symmetric BIB design whose existence can not be excluded by the Bruck Chowla Ryser Theorem? Note that the notion quasi-residual'' and quasi-derived'' are not essentially different, as pointed out by Van Lint and Tonchev [4], since replacing a symmetric design by its complement transforms its residual designs into derived designs and vice versa.