Cocyclic Hadamard Matrices Research Articles

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the concept of the pseudococycle of a partial Hadamard matrix over a Latin rectangle, whose fundamentals are comprehensively studied and illustrated.

On Cocyclic Hadamard Matrices over Goethals-Seidel Loops

About twenty-five years ago, Horadam and de Launey introduced the cocyclic development of designs, from which the notion of cocyclic Hadamard matrices developed over a group was readily derived. Much more recently, it has been proved that this notion may naturally be extended to define cocyclic Hadamard matrices developed over a loop. This paper delves into this last topic by introducing the concepts of coboundary, pseudocoboundary and pseudococycle over a quasigroup, and also the notion of the pseudococyclic Hadamard matrix. Furthermore, Goethals-Seidel loops are introduced as a family of Moufang loops so that every Hadamard matrix of Goethals-Seidel type (which is known not to be cocyclically developed over any group) is actually pseudococyclically developed over them. Finally, we also prove that, no matter if they are pseudococyclic matrices, the usual cocyclic Hadamard test is unexpectedly applicable.

Constructing cocyclic Hadamard matrices of order 4p

AbstractCocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order based on relative difference sets in groups of order ; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order with a prime; we prove refined structure results and provide a classification for . Our analysis shows that every CHM of order with is equivalent to a HM with one of five distinct block structures, including Williamson‐type and (transposed) Ito matrices. If , then every CHM of order is equivalent to a Williamson‐type or (transposed) Ito matrix.

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.

A Mixed Heuristic for Generating Cocyclic Hadamard Matrices

A way of generating cocyclic Hadamard matrices is described, which combines a new heuristic, coming from a novel notion of fitness, and a peculiar local search, defined as a constraint satisfaction problem. Calculations support the idea that finding a cocyclic Hadamard matrix of order $$4 \cdot 47$$ might be within reach, for the first time, progressing further upon the ideas explained in this work.

Gröbner bases and cocyclic Hadamard matrices

Hadamard ideals were introduced in 2006 as a set of nonlinear polynomial equations whose zeros are uniquely related to Hadamard matrices with one or two circulant cores of a given order. Based on this idea, the cocyclic Hadamard test enables us to describe a polynomial ideal that characterizes the set of cocyclic Hadamard matrices over a fixed finite group G of order 4t. Nevertheless, the complexity of the computation of the reduced Gröbner basis of this ideal is 2O(t2), which is excessive even for very small orders. In order to improve the efficiency of this polynomial method, we take advantage of some recent results on the inner structure of a cocyclic matrix to describe an alternative polynomial ideal that also characterizes the aforementioned set of cocyclic Hadamard matrices over G. The complexity of the computation decreases in this way to 2O(t). Particularly, we design two specific procedures for looking for Zt×Z22-cocyclic Hadamard matrices and D4t-cocyclic Hadamard matrices, so that larger cocyclic Hadamard matrices (up to t≤39) are explicitly obtained.

Hadamard full propelinear codes of type Q; rank and kernel

Hadamard full propelinear codes (HFP-codes) are introduced and their equivalence with Hadamard groups is proven (on the other hand, it is already known the equivalence of Hadamard groups with relative $(4n,2,4n,2n)$-difference sets in a group and also with cocyclic Hadamard matrices). We compute the available values for the rank and dimension of the kernel of HFP-codes of type Q and we show that the dimension of the kernel is always 1 or $2$. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an HFP-code of length $4n$, dimension of the kernel $k=2$, and maximum rank $r=2n$, we obtain an HFP-code of double length $8n$, dimension of the kernel $k=2$, and maximum rank $r=4n$.

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.

On ‐Cocyclic Hadamard Matrices

AbstractA characterization of ‐cocyclic Hadamard matrices is described, depending 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 to use and the way in which they have to be combined in order to obtain a ‐cocyclic Hadamard matrix. Exhaustive searches have been performed, so that the table in p. 132 in A. Baliga, K. J. Horadam, Australas. J. Combin., 11 (1995), 123–134 is corrected and completed. Furthermore, we identify four different operations on the set of coboundaries defining ‐cocyclic matrices, which preserve orthogonality. We split the set of Hadamard matrices into disjoint orbits, define 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 be the set of cocyclic Hadamard matrices over having a symmetric diagram. We also prove that the set of Williamson‐type matrices is a subset of of size .

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, $${\overline{B}(\mathsf{\textstyle Z\kern-0.4em Z}[G])}$$, 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 Alvarez 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 (Alvarez et al. in 2006).

Difference sets and doubly transitive actions on Hadamard matrices

Non-affine groups acting doubly transitively on a Hadamard matrix have been classified by Ito. Implicit in this work is a list of Hadamard matrices with non-affine doubly transitive automorphism group. We give this list explicitly, in the process settling an old research problem of Ito and Leon.We then use our classification to show that the only cocyclic Hadamard matrices developed from a difference set with non-affine automorphism group are those that arise from the Paley Hadamard matrices.If H is a cocyclic Hadamard matrix developed from a difference set then the automorphism group of H is doubly transitive. We classify all difference sets which give rise to Hadamard matrices with non-affine doubly transitive automorphism group. A key component of this is a complete list of difference sets corresponding to the Paley Hadamard matrices. As part of our classification we uncover a new triply infinite family of skew-Hadamard difference sets. To our knowledge, these are the first skew-Hadamard difference sets to be discovered in non-abelian p-groups with no exponent restriction.As one more application of our main classification, we show that Hallʼs sextic residue difference sets give rise to precisely one cocyclic Hadamard matrix.

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.

Hadamard matrices and their applications: Progress 2007–2010

We survey research progress in Hadamard matrices, especially cocyclic Hadamard matrices, their generalisations and applications, made over the past three years. Advances in 20 specific problems and several new research directions are outlined. Two new problems are presented.

On twin prime power Hadamard matrices

In this paper, we show that exactly one Hadamard matrix constructed using the twin prime power method is cocyclic. We achieve this by showing that the action of the automorphism group of a Hadamard matrix developed from a difference set induces a 2-transitive action on the rows of the matrix or is intransitive. We then use Ito's classification of Hadamard matrices with 2-transitive automorphism groups to derive a necessary condition on the order of a cocyclic Hadamard matrix developed from a difference set. This work answers a research problem posed by K.J. Horadam, and exhibits the first known infinite family of Hadamard matrices which are not cocyclic.

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.

The cocyclic Hadamard matrices of order less than 40

In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.

On the asymptotic existence of cocyclic Hadamard matrices

Let q be an odd natural number. We prove there is a cocyclic Hadamard matrix of order 2 10 + t q whenever t ⩾ 8 ⌊ log 2 ( q − 1 ) 10 ⌋ . We also show that if the binary expansion of q contains N ones, then there is a cocyclic Hadamard matrix of order 2 4 N − 2 q .

The homological reduction method for computing cocyclic Hadamard matrices

An alternate method for constructing (Hadamard) cocyclic matrices over a finite group G is described. Provided that a homological model B ̄ ( Z [ G ] ) ϕ : ⇌ H F h G for G is known, the homological reduction method automatically generates a full basis for 2-cocycles over G (including 2-coboundaries). From these data, either an exhaustive or a heuristic search for Hadamard cocyclic matrices is then developed. The knowledge of an explicit basis for 2-cocycles which includes 2-coboundaries is a key point for the designing of the heuristic search. It is worth noting that some Hadamard cocyclic matrices have been obtained over groups G for which the exhaustive searching techniques are not feasible. From the computational-cost point of view, even in the case that the calculation of such a homological model is also included, comparison with other methods in the literature shows that the homological reduction method drastically reduces the required computing time of the operations involved, so that even exhaustive searches succeeded at orders for which previous calculations could not be completed. With aid of an implementation of the method in Mathematica, some examples are discussed, including the case of very well-known groups (finite abelian groups, dihedral groups) for clarity.

Cocyclic Hadamard matrices from forms over finite Frobenius rings

We give a construction of cocyclic Butson–Hadamard matrices from bilinear forms over finite Frobenius rings, which generalizes previous constructions. We then prove a classification result, showing that the resulting matrices depend only on the additive group of the underlying module of the form.

A system of equations for describing cocyclic Hadamard matrices

AbstractGiven a basis ${\cal B} = \{f_1,\ldots, f_k\}$ for 2‐cocycles $f:G \times G \rightarrow \{\pm 1\}$ over a group G of order $\vert G\vert=4t$, we describe a nonlinear system of 4t‐1 equations and k indeterminates $x_i$ over ${\cal Z}_2, 1\leq i \leq k$, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that ($x_1,\ldots ,x_k$) is a solution of the system if and only if the 2‐cocycle $f=f_1^{x_1}\cdots f_k^{x_k}$ gives rise to a cocyclic Hadamard matrix $M_f=(f(g_i,g_j))$. Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in ${\cal B}$ which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups ${\cal Z}_2^2 \times {\cal Z}_t$ and $D_{4t}$. A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups $D_{4t}$, we have found that it suffices to check t instead of the 4t rows of $M_f$, to decide the Hadamard character of the matrix (for a special class of cocycles f). © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 276–290, 2008