A recursive method of A. C. Mukhopadhay is used to obtain several new infinite classes of Hadamard matrices. Unfortunately none of these constructions give previously unknown Hadamard matrices of order <40,000.

We consider the following problem arising in agricultural statistics. Suppose that a large number of plants are set out on a regular grid, which may be triangular, square or hexagonal, and that among these plants, half are to be given one and half the other of two possible treatments. For the sake of statistical balance, we require also that, if one plant in every k plants has i of its immediate neighbours receiving the same treatment as itself, then k is constant over all possible values of i. For square and triangular grids, there exist balanced arrays of finite period in each direction, but for the hexagonal grid, we show that no such balanced array can exist. Several related questions are discussed.

It is well known that in any (v, b, r, k, λ) resolvable balanced incomplete block design that b≧ ν + r − l with equality if and only if the design is affine resolvable. In this paper, we show that a similar inequality holds for resolvable regular pairwise balanced designs ((ρ, λ)-designs) and we characterize those designs for which equality holds. From this characterization, we deduce certain results about block intersections in (ρ, λ)-designs.

Commutative idempotent quasigroups with a sharply transitive automorphism group G are described in terms of so-called Room maps of G. Orthogonal Room maps and skew Room maps are used to construct Room squares and skew Room squares. Very general direct and recursive constructions for skew Room maps lead to the existence of skew Room maps of groups of order prime to 30. Also some nonexistence results are proved.

A series of balanced incomplete block (BIB) designs with parameters , is constructed, where , w any integer.

The structure is determined for the existence of some amicable weighing matrices. This is then used to prove the existence and non-existence of some amicable orthogonal designs in powers of two.

An equidistant permutation array (EPA) is a ν × r array defined on an r-set, R, such that (i) each row is a permutation of the elements of R and (ii) any two distinct rows agree in λ positions (that is, the Hamming distance is (r−λ)).

Such an array is said to have order ν. In this paper we give several recursive constructions for EPA's.

The first construction uses a resolvable regular pairwise balanced design of order v to construct an EPA of order ν. The second construction is a generalization of the direct product construction for Room squares.

We also give a construction for intersection permutation arrays, which arrays are a generalization of EPA's.

For some integer modulus q let F be a subset of Z(q), the additive group of residues modulo q. As in «1» we shall write

and

In 1944 and 1945 H. Hadwiger [1, 2] proved the following theorems.

Theorem A. Let En be covered by n + 1 closed sets. Then there is one of the sets, within which all distances are realized.

Theorem B. Let En be covered by 4n−3 closed sets that are all mutually congruent. Then all distances are realized within each set.

Here a distance d is realized within a set S, if there are points x, y in S at distance d apart.

The purpose of this paper is to demonstrate that a number of properties of independence spaces are of finite character, thus making it possible to easily generalise known theorems for finite spaces, or matroids, to independence spaces on infinite sets.