On ZZt × ZZ2 2-cocyclic Hadamard matrices

A characterization of ZZt × ZZ22 -cocyclic Hadamard matrices is described, de- pending on the notions of distributions, ingredients and recipes. In particular, these notions lead to the establishment of some bounds on the number and distribution of 2-coboundaries over ZZt × ZZ22 to use and the way in which they have to be combined in order to obtain a ZZt × ZZ22 -cocyclic Hadamard matrix. Exhaustive searches have been performed, so that the table in p. 132 in [4] is corrected and completed. Furthermore, we identify four different operations on the set of coboundaries defining ZZt × ZZ22 -cocyclic matrices, which preserve orthogonality. We split the set of Hadamard matrices into disjoint orbits, de- fine representatives for them and take advantage of this fact to compute them in an easier way than the usual purely exhaustive way, in terms of diagrams. Let H be the set of cocyclic Hadamard matrices over ZZt × ZZ22 having a sym- metric diagram. We also prove that the set of Williamson type matrices is a subset of H of size |H| t .Junta de Andalucía FQM-01

A Generalised Hadamard Transform

A Generalised Hadamard Transform for multi-phase or multilevel signals is introduced, which includes the Fourier, Generalised, Discrete Fourier, Walsh-Hadamard and Reverse Jacket Transforms. The jacket construction is formalised and shown to admit a tensor product decomposition. Primary matrices under this decomposition are identified. New examples of primary jacket matrices of orders 8 and 12 are presented.Comment: To appear in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200

Direct sums of balanced functions, perfect nonlinear functions, and orthogonal cocycles

Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non-cyclic abelian groups and use it to find all the orthogonal cocycles over Z2t, 2 t 4. We conjecture that any orthogonal cocycle over Z2t, t 2, must be multiplicative

A polynomial approach to cocycles over elementary abelian groups

We derive bivariate polynomial formulae for cocycles and coboundaries in Z2(xs2124pn,xs2124pn), and a basis for the (pn-1-n)-dimensional GF(pn)-space of coboundaries. When p=2 we determine a basis for the $(2^n + {n\choose 2} -1)$-dimensional GF(2n)-space of cocycles and show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form

Secure optical layer flexibility in 5G networks

We propose an adaptive resource allocation framework for on-demand communications in a software-defined mobile fronthaul (MFH) network that supports dynamic processing resource sharing. Our theoretical and experimental studies point to the feasibility of secure bidirectional transmission with guaranteed bit error rate (BER) service using adaptive modulation and coding.This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Equivalences of Zt×Z22-cocyclic Hadamard matrices

One of the most promising structural approaches to resolving the Hadamard Conjecture uses the family of cocyclic matrices over Zt × Z2 2. Two types of equivalence relations for classifying cocyclic matrices over Zt × Z2 2 have been found. Any cocyclic matrix equivalent by either of these relations to a Hadamard matrix will also be Hadamard. One type, based on algebraic relations between cocycles over any fi- nite group, has been known for some time. Recently, and independently, a second type, based on four geometric relations between diagrammatic visualisations of cocyclic matrices over Zt × Z2 2, has been found. Here we translate the algebraic equivalences to diagrammatic equivalences and show one of the diagrammatic equivalences cannot be obtained this way. This additional equivalence is shown to be the geometric translation of matrix transposition

A simple construction of complex equiangular lines

A set of vectors of equal norm in $\mathbb{C}^d$ represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in $\mathbb{C}^d$ for every $d \geq 2$. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.Comment: 11 pages; minor revisions and comments added in section 1 describing a link to previously known results; correction to Theorem 1 and updates to reference

The homology of groupnets

In this thesis I use the theory of groupnets (Brandt groupoids) to investigate the homology of mapping cylinder groupnets; that is, groupnets G which are the homotopy colimits of diagrams (V, A) of groupnets. When the edge morphisms of (V, A) are all monomorphisms, G is known as a graph product. The principal result of the thesis is the construction of a G-complex with universal properties - the G-mapping cylinder - from a diagram of complexes corresponding to (V, A) , and the subsequent proof that if G is a graph product and the vertex complexes are all free resolutions of their respective trivial modules, then the G-mapping cylinder is a free resolution of its trivial module . An extension of the categorical approach to rings and modules is developed in order to provide the general result. The notion of chain homotopy is also extended to a form strongly motivated by the topological definition of homotopy. The mapping cylinder complex determines MayerVietoris sequences for the homology of graph products, which in turn may be used to extend several results on duality groups. For each group in a certain class of groupnets with cohomological dimension two (including torsion-free one-relater groups and tree products of free groups ), the mapping cylinder may be employed to evaluate a comultiplication which gives a coring structure to the integral homology module of the group. This comultiplication is in turn analysed (though not in full generality) to provide further information about the group