In this paper, the notion of trades over finite fields is introduced. In particular, trades over GF (3) (ternary trades) are studied. By considering the incidence matrix of t-subsets vs. k-subsets of a v-set as a parity check matrix of a ternary code, we obtain a new family of codes in which every codeword is a ternary trade. The spectrum of weights of these codes is discussed; a simple and fas...

A k-bowtie is the graph consisting of two cycles of lenght k, having exactly one vertex in common. A k-bowtie system of order v is a pair (X,B), where X is a finite set of v vertices and B is a collection of edge disjoint k-bowties (called blocks) which partitions the edge set of the complete graph Kn, defined in X. In this paper we study 4-bowtie systems. We determine the spectrum of all types...

A program is outlined for the enumeration of unital 2-(28,4,1) designs that uses tactical decompositions de1ned by vectors of certain weight in the dual binary code of a design. A class of designs with a spread that covers a codeword of weight 12 is studied in detail. A total of 909 nonisomorphic designs are constructed that include the classical hermitian and Ree unitals, as well as many other...

A group divisible design GDD(m,n; 3, λ1, λ2) is an ordered triple (V,G,B), where V is a m+ n-set of symbols, G is a partition of V into 2 sets of sizes m,n, each set being called group, and B is a collection of 3-subsets (called blocks) of V , such that each pair of symbols from the same group occurs in exactly λ1 blocks; and each pair of symbols from different groups occurs in exactly λ2 block...

A three-fold triple system is a design wherein each pair of treatments occurs exactly once. One way to construct this design is by using an idempotent commutative quasigroup. This paper attempts to provide another method of constructing a 3-fold triple system. Firstly, we would like to discuss compatible factorization without multiple edges using a patterned starter construction. Then, we will ...

A 5-cycle system on v + w points embeds a balanced P4-design on v points if there is a subset of v points on which the 5-cycles induce the blocks of a balanced P4-design. The mininum possible such w is: v (mod 30) w v ≡ 4, 16 v−1 3 v ≡ 7, 10, 22, 25 v+5 3 v ≡ 1, 13, 28 v+11 3 v ≡ 19 (mod 30) v+17 3 We shall show this minimum value of w is attained. AMS classification: 05B05.

For every positive integer m, we construct a symmetric (v, k, λ)-design with parameters v = h((2h−1) 2m−1) h−1 , k = h(2h − 1)2m−1, and λ = h(h − 1)(2h − 1)2m−2, where h = ±3 · 2 and |2h − 1| is a prime power. For m ≥ 2 and d ≥ 1, these parameter values were previously undecided. The tools used in the construction are balanced generalized weighing matrices and regular Hadamard matrices of order...

Let B(k, λ) be the spectrum of integers n such that there exists a Sλ(2, k, n), a balanced incomplete block design of order n, block size k and index λ. Lindner and Rosa [6] introduced the definition of a Sλ(2, 4, n) having a metamorphosis into a Sλ(2, 3, n) and proved that the necessary condition n ∈ B(3, λ) ∩B(4, λ) is also sufficient. The aim of this paper is to present two different general...

For any positive integers k¿ 3 and , let cd(k; ) denote the smallest integer such that the necessary conditions 2 (v − 1) ≡ 0 (mod k − 1) and v(v − 1) ≡ 0 (mod ( k2 )) for the existence of a DB(k; ; v) are also su3cient for every v¿ cd(k; ). In this article we provide an estimate for cd(k; ) when k ≡ 0 (mod 4) and any . Combined with the results in (Discrete Math. 222 (2000) 27–40), we complete...

All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of inequivalent Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13,680,757 Hadamard matrices of one type and 26,369 such matrices of another t...