Embedding cocylic D-optimal designs in cocylic Hadamard matrices

A method for embedding cocyclic submatrices with “large” determinants of orders 2t in certain cocyclic Hadamard matrices of orders 4t is described (t an odd integer). If these determinants attain the largest possible value, we are embedding D-optimal designs. Applications to the pivot values that appear when Gaussian elimination with complete pivoting is performed on these cocyclic Hadamard matrices are studied.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM-016Junta de Andalucía P07-FQM-0298

Orthogonal Arrays and Legendre Pairs

Well-designed experiments greatly improve test and evaluation. Efficient experiments reduce the cost and time of running tests while improving the quality of the information obtained. Orthogonal Arrays (OAs) and Hadamard matrices are used as designed experiments to glean as much information as possible about a process with limited resources. However, constructing OAs and Hadamard matrices in general is a very difficult problem. Finding Legendre pairs (LPs) results in the construction of Hadamard matrices. This research studies the classification problem of OAs and the existence problem of LPs. In doing so, it makes two contributions to the discipline. First, it improves upon previous classification results of 2-symbol OAs of even-strength t and t+2 columns. Second, it presents previously unknown impossible values for the dimension of the convex hull of all feasible points to the LP problem improving our understanding of its feasible set

Some remarks on Hadamard matrices

In this note we use combinatorial methods to show that the unique,upto equivalence, $5 \times 5$ submatrix of elements $\pm 1$ with determinant 48, the unique, upto equivalence, $6 \times 6$ submatrix of elements $\pm 1$ with determinant 160 and the unique, upto equivalence, $7 \times 7$ submatrix of elements $\pm 1$ with determinant 576 cannot be embedded in the Hadamard matrix of order 8. Some properties of Sylvester Hadamard matrices, their Smith Normal Forms and pivot patterns of Hadamard matrices when Gaussian Elimination with complete pivoting is applied on them, are also reviewed. Using the appearing pivot values, we reconfirm the above non embedding results

Some remarks on Hadamard matrices

In this note we use combinatorial methods to show that the unique, up to equivalence, 5 ×5 (1, - 1)-matrix with determinant 48, the unique, up to equivalence, 6 ×6 (1, - 1)-matrix with determinant 160, and the unique, up to equivalence, 7 ×7 (1, - 1)-matrix with determinant 576, all cannot be embedded in the Hadamard matrix of order 8. We also review some properties of Sylvester Hadamard matrices, their Smith Normal Forms, and pivot patterns of Hadamard matrices when Gaussian Elimination with complete pivoting is applied on them. The pivot values which appear reconfirm the above non-embedding results. © 2010 Springer Science + Business Media, LLC