Some Properties of Perfect Matroid Designs

Blocks of a matroid are called hyperplanes. For various definitions and results connected with matroids, see [26]. Subsets of X , which are intersections of hyperplanes are called flats of a matroid. Each subset Y c_ X has a well-defined rank. If F is a flat of rank i and x e X \ F , then, there is a unique flat of rank (i + 1) which contains FU{x}. Rank of X is said to be the rank of matroid. Let M be a matroid of rank r. We will denote by Mi the matroid Mi = ( X , pi), where pi is the set of all i-flats (i.e., flats of rank i) of M, l s i s r . M, is called (r-i)th truncation of M. We will use the usual geometric terminology and call 1-flats points, 2-flats lines, etc. A perfect matroid design (PMD) is a matroid of rank r, such that, for any integer i, the cardinality of all i-flats is the same number ai, 0 < i s r. For simplicity we consider only simple PMDs, i.e., PMD for which a,, = 0 and a1 = 1. We do not lose much generality by this assumption (see Section 2). We will write a2 = I , c ~ + ~ = k and a ,=IX(=u . The set a={a,=0, a ,= l , az=l, aj , . . . , a , , = k ,