A structure for implementing lapped transforms with time-varying block sizes is presented which allows full orthogonality of the transient transforms. The formulation is based on a factorization of the transfer matrix into orthogonal factors. Such an approach can be viewed as a sequence of stages with variableblock-size transforms separated by sample-shu ing (delay) stages. Details and design e...

Balanced nested designs are closely related to other combinatorial structures such as balanced arrays and balanced n-ary designs. In particular, the existence of symmetric balanced nested designs is equivalent to the existence of some balanced arrays. In this paper, various constructions for symmetric balanced nested designs are provided. They are used to determine the spectrum of symmetric bal...

An (v, •, 2) packing design of order v, block size ~, and index 2 is a collection of ~:-element subsets, called blocks, of a set V such that every 2-subsets of V occurs in at most 2 blocks. The packing problem is to determine the maximum number of blocks in a packing design. In this paper we provide a powerful technique for constructing designs and solve the packing problem in the case tc = 5, ...

We study three similar bijections on set partitions. The first gives a bijective proof of the equivalence of two statistics with a q-Stirling distribution, Milne’s statistic and the intertwining number. The second proves the equivalence of a multivariate block size distribution to a covering statistic. The third demonstrates equivalence of the number of all set partitions up to a given size to ...

Let V be a finite set of order v. A (V,K,A) packing design of index A and block size IC is a collection of K-element subsets, called blocks, such that every 2-subset of V occurs in at most A blocks. The packing problem is to determine the maximum number of blocks, D(V,K,A), in a packing design. It is well known that D(V,K,A) S [-i [:=~ ~]] : : t(VIK,~), where [xl is the largest integer satisfyi...

In this note the existence of a (Vj P2j 4, 2) BTD I for P2 = 0, 1 and 2, in which any pair of blocks intersect in at most two elements, is proved for any admissible v. A balanced ternary design is a collection of multi-sets of size k, chosen from a v-set in such a way that each element occurs 0, 1 or 2 times in anyone block, each pair of non-distinct elements, {x, x}, occurs in P2 blocks of the...

Let G be the projective special linear group PSL(2, 2), let X be the projective line and B be any subgroup of GF ∗(2n). We give a new infinite family of simple 3-designs by determining the parameter set of (X, G(B0)), where B0 = B ∪ {0}.

We find a natural construction of a large class of symmetric graphs from pointand block-transitive 1-designs. The graphs in this class can be characterized as G-symmetric graphs whose vertex sets admit a G-invariant partition B of block size at least 3 such that, for any two blocks B,C of B, either there is no edge between B and C , or there exists only one vertex in B not adjacent to any verte...

A Mendelsohn design .M D(v, k,).) is a pair (X, B), where X is a vset together with a collection B of ordered k-tuples from X such that each ordered pair from X is contained in exactly ). k-tuples of B. An M D(v, k,).) is said to be self-converse, denoted by SC!'vf D( v, k,).) = (X,B,/), if there is an isomorphism / from (X, B) to (X,B), where B-1 {(Xk,:r:k-l, ... ,X2,Xl); (Xl,,,,,Xk) E B}. The...

A necessary condition is given for the existence of some Generalised Bhaskar Rao designs (GBRDs) with odd block size over cyclic groups of even order. Some constructions are given for GBRDs over cyclic groups of even order with block size 3 and with block size 4. AMS Subject Classification: 05B99 J( ey words and phrases: Balanced Incomplete Block Designs; Generalised Bhaskar Rao Designs