7,074 research outputs found
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
On minors of maximal determinant matrices
By an old result of Cohn (1965), a Hadamard matrix of order n has no proper
Hadamard submatrices of order m > n/2. We generalise this result to maximal
determinant submatrices of Hadamard matrices, and show that an interval of
length asymptotically equal to n/2 is excluded from the allowable orders. We
make a conjecture regarding a lower bound for sums of squares of minors of
maximal determinant matrices, and give evidence in support of the conjecture.
We give tables of the values taken by the minors of all maximal determinant
matrices of orders up to and including 21 and make some observations on the
data. Finally, we describe the algorithms that were used to compute the tables.Comment: 35 pages, 43 tables, added reference to Cohn in v
On permanents of Sylvester Hadamard matrices
It is well-known that the evaluation of the permanent of an arbitrary
-matrix is a formidable problem. Ryser's formula is one of the fastest
known general algorithms for computing permanents. In this paper, Ryser's
formula has been rewritten for the special case of Sylvester Hadamard matrices
by using their cocyclic construction. The rewritten formula presents an
important reduction in the number of sets of distinct rows of the matrix to
be considered. However, the algorithm needs a preprocessing part which remains
time-consuming in general
- …