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For a v -set X , W 23 ( v ) is a ( v 2 ) × ( v 3 ) inclusion matrix where rows and columns are indexed by pairs and triples of X , respectively, and for row T and column K , W 23 ( v ) ( T , K ) = 1 if T ⊆ K and zero otherwise. In this paper, we classify the basis elements of the null Z ( W 23 ( v ) ) , derived from the Gaussian elimination on W 23 ( v ) (called standard basis), into five classes. Then, we present a new algorithm to construct a ( 2 , 3 , v ) -halving for a feasible v , i.e. a nowhere zero T ( 2 , 3 , v ) -trade. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper we focus our research on calculating minors of weighing matrices of order n and weight n − k, denoted by W(n, n − k). We provide analytical determinant computations, counting techniques for specifying the existence of certain submatrices inside a W(n, n − k) and an algorithm for computing the (n − j)×(n − j) minors of a W(n, n − k), which is realized with the notion of symbolic manipulation. These results are valid of general n. The ideas presented in this work can be used as the fundamental basis, on which the calculation of minors of other weighing matrices, and in general of orthogonal matrices, can be developed. An application of the derived formulas to an interesting problem of Numerical Analysis, the growth problem, is also presented. [Copyright &y& Elsevier]

Abstract: Koukouvinos et al. [C. Koukouvinos, M. Mitrouli, J. Seberry, Growth in Gaussian elimination for weighing matrices, W(n, n −1), Linear Algebra Appl. 306 (2000) 189–202], conjectured that the growth factor for Gaussian elimination of any completely pivoted weighing matrix of order n and weight n −1 is n −1 and that the first and last few pivots are for n >14. In the present paper we study the growth problem for skew and symmetric conference matrices. An algorithm for extending a k × k matrix with elements 0, ±1 to a skew and symmetric conference matrix of order n is described. By using this algorithm we show that the unique W(8,7) has two pivot structures. We also prove that the unique W(10,9) has three pivot patterns. [Copyright &y& Elsevier]