Consider an incidence structure whose points are the points of a PG n (n + 2, q) and whose block are the subspaces of codimension two, where n 2. Since every two subspaces of codimension two intersect in a subspace of codimension three or codimension four, it is easily seen that this incidence structure is a quasi-symmetric design. The aim of this paper is to prove a characterization of such designs (that are constructed using projective geometries) among the class of all the quasi-symmetric designs with correct parameters and with every block a good block. The paper also improves an earlier result for the special case of n = 2 and obtains a Dembowski-Wagner-type result for the class of all such quasi-symmetric designs.A block design D with parameters (v, b, r, k, ) is called a quasi-symmetric design if every pair of blocks of D intersect in x or y points, where, by convention, we assume that x < y. The definition of a good block for general block designs is in the literature (see, e.g. [1]). Quasi-symmetric designs with good blocks were studied by McDonough and Mavron [4] when x = 0 and their results have been recently extended by Mavron et al. in [3] for the case when x = 1.Definition. Let D be a quasi-symmetric design and X a block of D. Then X is called a good block if the following condition is satisfied: given any block Y such that |Y ∩ X| = y and given any point p not contained in X ∪ Y , there exists a (unique) block Z such that Z contains both X ∩ Y and p.By a PG n (m, q), we mean a design (that is a 2-design) which is obtained by taking as points, the points of the projective geometry PG(m, q) (of dimension m over a field with q elements) and whose blocks are all the n-dimensional subspaces of PG(m, q). It is well-known that this is a 2-design for all 1 n < m. Further, if we take m = n + 2, then the blocks of our design are n-dimensional subspaces of PG(n + 2, q) and hence every two blocks intersect in either a subspace of dimension n − 1 or n − 2 and thus the design we obtain is a quasi-symmetric design. The aim of this paper is to prove the following characterization theorem.
An extension theorem for t-designs is proved. As an application, a class of 4-(4" + 1,5,2) designs is constructed by extending designs related to the 3-designs formed by the minimum weight vectors in the Preparata code of length n = 4", m 2 2. 0 1994 John Wiley & Sons, Inc.
. INTRODUCTIONWe assume familiarity with some basic notions from design theory and coding theoryis a collection B of (not necessarily distinct) k-subsets (called blocks) of a v-set X (with elements called points) such that every t-subset of X is contained in precisely h blocks. A design with h = 1 is called a Steiner system, and the notation S(t, k , v) is often used in this case. Any t -(v, k , A) design is also an s-design for s 5 t with parameters s-(v, k , As), where (Cf.7 e.g.9 PI, [41, t71).
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