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The quantity g (k) (v) was introduced in [6] as the minimum number of blocks necessary in a pairwise balanced design on v elements, subject to the condition that the longest block has length k. Recently, we have needed to use all possibilities for such minimal covering designs, and we record all non-isomorphic solutions to the problem for v ≤ 13.

The minimum number of incomplete blocks required to cover, exactly λ times, all t-element subsets from a set V of cardinality v (v > t) is denoted by g(λ, t; v).

- Leonard Baumert, S W Golomb, Marshall Hall, Jr, J Williamson, A Brauer +4 others
- 2007

An Hadamard matrix H is an n by n matrix all of whose entries are +1 or — 1 which satisfies HH T = n J, H T being the transpose of H. The order n is necessarily 1, 2 or 42, with t a positive integer. R. E. A. C. Paley [3] gave construction methods for various infinite classes of Hadamard matrices, chiefly using properties of quadratic residues in finite… (More)

- R. G. Stanton
- 2003

A bstract The classical bicovering problem seeks to cover all pairs from a v-set by a family F of k-sets so that every pair occurs at least twice and the cardinality of F is minimaL A weight function is introduced for blocks in such a design, and its use in constructing bicoverings is illustrated. A covering is a collection of k-sets (blocks) chosen from… (More)

A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A (κ, τ)-regular set is a subset of the vertices inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbours… (More)

A Costas latin square of order n is a set of n disjoint Costas arrays of the same order. Costas latin squares are studied here from a construction as well as a classification point of view. A complete classification is carried out up to order 27. In this range, we verify the conjecture that there is no Costas latin square for any odd order n ≥ 3. Various… (More)

This is a preprint of an article accepted for publication in the Ars Combina-toria c 2004 (copyright owner as specified in the journal). Abstract We survey the status of minimal coverings of pairs with block sizes two, three and four when λ = 1, that is, all pairs from a v-set are covered exactly once. Then we provide a complete solution for the case λ = 2.

The minimum number of blocks having maximum size precisely five that is required to cover, exactly once, all pairs of elements from a set of cardinality v is denoted by g (5)