We study several classes of arrays generalizing designs and orthogonal arrays. We use these to construct non-trivial t-designs without repeated blocks for all t.
1. Designs and arrays
In this paper, we will assume all sets that are not obviously infinite, to be finite. If X and Y are sets, X Y denotes the set of all functions from Y to X. If X is a set, then ~(X) is the set of all subsets of X and ~t(X), t e N, the set of all t-subsets of X. If Y ~-X, ~PY.X e {0, 1} x denotes the characteristic function. If X is a set, ix is the identity function on X. A t-X-multiset will be a function/z :X--> N such that I/z[ = Y,x~xlt(x) = t. We call/~(x) the multiplicity ofx. We call x an element of/z if /z(x) ~ 0 and a repeated element if/z(x) t> 2. If ~ has no repeated elements, i.e., if /u(X) c {0, 1}, then /z = ~/~-1(1), X and /z can be identified with /t-x(1)cX. If X1 ~ X2, we identify Nx, with the subset of N x~ consisting of all /t such that l(x2-x0 = 0. A t-design S(A; t, k, v), )., t, k, v ~ N, t ~< k, is a ~k(S)-multiset ~, ISI = v, such that for every T e ~,(S), we have Er~-B~*~s)/u(B)= ).. The elements of/t are called blocks. It has been known for a long time that there are a lot of t-designs for all t (see for instance or ). However, until very recently, the only known t-designs without repeated blocks and with t~ > 6 were the complete designs ISI = v, k I> t, which are S((~-~); t, k, v). Magliveras and Leavitt [5] constructed six non-isomorphic S(36;6, 8, 33) without repeated blocks and Kramer, Leavitt and Magliveras constructed two non-isomorphic S(112; 6, 9, 20) without repeated blocks. In this paper, we will construct infinitely many non-trivial t-designs without repeated blocks for all t.
If S and J are sets, a (J, S)-array or simply array will be an S~-multiset. The elements of/~ are called rows. The elements of J are called columns. We identify A = S a with ~a,s' and B e S ~ with {B} and so with ~B~.s~. Thus we can apply all definitions given for arrays to subsets of S J and dements of S I. If /~ is a (J,S)-array, ff Ol:J-'-~J1 and o2:S---~$1 are bijections, we will denote by (ol, tr2)/~ the (J1, S1)-array defined by ((ol, o2)~)(B) = ~(O2 lยฐ B o 01), B e S~ ~. If /zl is a (-/1, S1)-array, we call (01, tr2) a bi-isomorphism between /z and /~1 if /~1 = (trl, tr2)/z or, equivalently, if l~l(02oB o trl 1) =/~(B) for every B e S J. We call