27 research outputs found
On higher dimensional cocyclic Hadamard matrices
Provided that a cohomological model for G is known, we describe a method for constructing a basis for n-cocycles over G, from which the whole set of n-dimensional n-cocyclic matrices over G may be straightforwardly calculated. Focusing in the case n=2 (which is of special interest, e.g. for looking for cocyclic Hadamard matrices), this method provides a basis for 2-cocycles in such a way that representative 2-cocycles are calculated all at once, so that there is no need to distinguish between inflation and transgression 2-cocycles (as it has traditionally been the case until now). When n>2, this method provides an uniform way of looking for higher dimensional n-cocyclic Hadamard matrices for the first time. We illustrate the method with some examples, for n=2,3. In particular, we give some examples of improper 3-dimensional 3-cocyclic Hadamard matrices
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
The cohomological reduction method for computing n-dimensional cocyclic matrices
Provided that a cohomological model for is known, we describe a method
for constructing a basis for -cocycles over , from which the whole set of
-dimensional -cocyclic matrices over may be straightforwardly
calculated. Focusing in the case (which is of special interest, e.g. for
looking for cocyclic Hadamard matrices), this method provides a basis for
2-cocycles in such a way that representative -cocycles are calculated all at
once, so that there is no need to distinguish between inflation and
transgression 2-cocycles (as it has traditionally been the case until now).
When , this method provides an uniform way of looking for higher
dimensional -cocyclic Hadamard matrices for the first time. We illustrate
the method with some examples, for . In particular, we give some
examples of improper 3-dimensional -cocyclic Hadamard matrices.Comment: 17 pages, 0 figure
Cocyclic Hadamard matrices over Latin rectangles
In the literature, the theory of cocyclic Hadamard matrices has always been developed over finite groups.
This paper introduces the natural generalization of this theory to be developed over Latin rectangles. In
this regard, once we introduce the concept of binary cocycle over a given Latin rectangle, we expose
examples of Hadamard matrices that are not cocyclic over finite groups but they are over Latin rectangles.
Since it is also shown that not every Hadamard matrix is cocyclic over a Latin rectangle, we focus on
answering both problems of existence of Hadamard matrices that are cocyclic over a given Latin rectangle and
also its reciprocal, that is, the existence of Latin rectangles over which a given Hadamard matrix is cocyclic. We
prove in particular that every Latin square over which a Hadamard matrix is cocyclic must be the multiplication
table of a loop (not necessarily associative). Besides, we prove the existence of cocyclic Hadamard matrices
over non-associative loops of order 2t+3, for all positive integer t > 0.Junta de Andalucía FQM-01
Homological models for semidirect products of finitely generated Abelian groups
Let G be a semidirect product of finitely generated Abelian groups. We provide a method for constructing an explicit contraction (special homotopy equivalence) from the reduced bar construction of the group ring of G, B¯¯¯¯(ZZ[G]) , to a much smaller DGA-module hG. Such a contraction is called a homological model for G and is used as the input datum in the methods described in Álvarez et al. (J Symb Comput 44:558–570, 2009; 2012) for calculating a generating set for representative 2-cocycles and n-cocycles over G, respectively. These computations have led to the finding of new cocyclic Hadamard matrices (Álvarez et al. in 2006)
An algorithm for computing cocyclic matrices developed over some semidirect products
An algorithm for calculating a set ofgenerators ofrepresentative 2-cocycles on semidirect product offinite abelian groups is constructed, in light ofthe theory over cocyclic matrices developed by Horadam and de Launey in [7],[8]. The method involves some homological perturbation techniques [3],[1], in the homological correspondent to the work which Grabmeier and Lambe described in [12] from the viewpoint ofcohomology . Examples ofexplicit computations over all dihedral groups D 4t are given, with aid of Mathematica
Cubes of symmetric designs
We study -dimensional matrices with -entries (-cubes) such
that all their -dimensional slices are incidence matrices of symmetric
designs. A known construction of these objects obtained from difference sets is
generalized so that the resulting -cubes may have inequivalent slices. For
suitable parameters, they can be transformed into -dimensional Hadamard
matrices with this property. In contrast, previously known constructions of
-dimensional designs all give examples with equivalent slices.Comment: 18 page