48 research outputs found
On an inequivalence criterion for cocyclic Hadamard matrices
Given two Hadamard matrices of the same order, it can be quite difficult
to decide whether or not they are equivalent. There are some criteria to determine
Hadamard inequivalence. Among them, one of the most commonly used is the
4-profile criterion. In this paper, a reformulation of this criterion in the cocyclic
framework is given. The improvements obtained in the computation of the 4-profile
of a cocyclic Hadamard matrix are indicated.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM–296Junta de Andalucía P07-FQM-0298
Boolean Functions and Permanents of Sylvester Hadamard Matrices
One of the fastest known general techniques for computing permanents is Ryser’s formula. On this note, we show that this formula over Sylvester Hadamard matrices of order 2m, Hm, can be carried out by enumerating m-variable Boolean functions with an arbitrary Walsh spectrum. As a consequence, the quotient per(Hm)/22m might be a measure of the “density” of m-variable Boolean functions with high nonlinearity
Generalized binary arrays from quasi-orthogonal cocycles
Generalized perfect binary arrays (GPBAs) were used by Jedwab to
construct perfect binary arrays. A non-trivial GPBA can exist only if its energy
is 2 or a multiple of 4. This paper introduces generalized optimal binary arrays
(GOBAs) with even energy not divisible by 4, as analogs of GPBAs. We give a
procedure to construct GOBAs based on a characterization of the arrays in terms
of 2-cocycles. As a further application, we determine negaperiodic Golay pairs
arising from generalized optimal binary sequences of small length.Junta de Andalucía FQM-01
On quasi-orthogonal cocycles
We introduce the notion of quasi-orthogonal cocycle. This
is motivated in part by the maximal determinant problem for square
{±1}-matrices of size congruent to 2 modulo 4. Quasi-orthogonal cocycles
are analogous to the orthogonal cocycles of algebraic design theory.
Equivalences with new and known combinatorial objects afforded by this
analogy, such as quasi-Hadamard groups, relative quasi-difference sets,
and certain partially balanced incomplete block designs, are proved.Junta de Andalucía FQM-01
Self-Dual codes from (−1,1)-matrices of skew symmetric type
Previously, self-dual codes have been constructed from weighing matrices,
and in particular from conference matrices (skew and symmetric). In this paper, codes
constructed from matrices of skew symmetric type whose determinants reach the Ehlich-
Wojtas’ bound are presented. A necessary and sufficient condition for these codes to be
self-dual is given, and examples are provided for lengths up to 52.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM-016Junta de Andalucía P07-FQM-0298
Almost supplementary difference sets and quaternary sequences with optimal autocorrelation
We introduce almost supplementary difference sets (ASDS). For odd
m, certain ASDS in Zm that have amicable incidence matrices are equivalent to
quaternary sequences of odd length m with optimal autocorrelation. As one consequence,
if 2m − 1 is a prime power, or m 1 mod 4 is prime, then ASDS of
this kind exist. We also explore connections to optimal binary sequences and group
cohomology.Junta de Andalucía FQM-01
Quasi-Hadamard Full Propelinear Codes
In this paper, we give a characterization of quasi-Hadamard groups in terms of propelinear codes. We
define a new class of codes that we call quasi-Hadamard full propelinear codes. Some structural properties of
these codes are studied and examples are provided.Junta de Andalucía FQM-016Ministerio de Economía y Competitividad TIN2016-77918-
On the computability of the p-local homology of twisted cartesian products of Eilenberg-Mac Lane spaces
Working in the framework of the Simplicial Topology, a method for calculating
the p-local homology of a twisted cartesian product X( , m, , 0, n) =
K( ,m)× K( 0, n) of Eilenberg-Mac Lane spaces is given. The chief technique
is the construction of an explicit homotopy equivalence between the normalized
chain complex of X and a free DGA-module of finite type M, via homological
perturbation. If X is a commutative simplicial group (being its inner product
the natural one of the cartesian product of K( ,m) and K( 0, n)), then M is a
DGA-algebra. Finally, in the special case K( , 1) ,! X
p!
K( 0, n), we prove
that M can be a small twisted tensor product
Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach
An n by n skew-symmetric type (−1, 1)-matrix K = [ki,j ] has 1’s on the main
diagonal and ±1’s elsewhere with ki,j = −kj,i. The largest possible determinant of such
a matrix K is an interesting problem. The literature is extensive for n 0 mod 4 (skew-
Hadamard matrices), but for n 2 mod 4 there are few results known for this question.
In this paper we approach this problem constructing cocyclic matrices over the dihedral
group of 2t elements, for t odd, which are equivalent to (−1, 1)-matrices of skew type.
Some explicit calculations have been done up to t = 11. To our knowledge, the upper
bounds on the maximal determinant in orders 18 and 22 have been improved.Junta de Andalucía FQM-01
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