On Cocyclic Hadamard Matrices over Goethals-Seidel Loops

Abstract

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.

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

“…In this section, we introduce the notions of both the pseudocoboundary and pseudococycle over a Latin rectangle as a natural generalization of the similar concepts described over quasigroups in [17] by keeping in mind, to this end, the concepts introduced in [18]. Firstly, let us define the types of Latin rectangles where such a generalization is feasible.…”

“…. When we want to refer to any h-pseudocoboundary (matrix) over L, we omit the prefix h. As such, the concept of the pseudocoboundary over a Latin rectangle constitutes a generalization of that over a quasigroup [17], which arises when r = n. In any case, the following result establishes that the pseudococyclic framework over Latin rectangles is not included in the cocyclic framework over such arrays. Hence, it constitutes a new proposal that has to be independently studied.…”

“…Finally, the following example enables us to ensure the existence of Hadamard matrices that are not cocyclic over any Latin rectangle, but that are pseudocoboundary matrices over a Latin square. Let us finish this section by introducing the notion of a pseudococycle over a Latin rectangle as a generalization of both the concepts of a cocycle over a Latin rectangle [18] and a pseudococycle over a quasigroup [17]. To this end, we take into account the previously described notion of a pseudocoboundary over Latin rectangles.…”

“…Very recently, a new approach introduced by the authors of [17] has successfully dealt with a cocyclic development of Goethals-Seidel arrays, not over a group, but over a family of Moufang loops. This approach is comprehended in the new theory of cocyclic development over quasigroups and Latin rectangles, which has also recently been introduced by the authors in [18].…”

“…The main aspect of this new approach is the fact that associativity is no longer a necessary condition for dealing with any of the concepts and results that are usually involved in the cocyclic development over finite groups. It is so thatthe existence of coboundaries over non-associative loops has already been proved [17]. In this regard, we remind the reader that a cocycle φ over a quasigroup (Q, •) is called a coboundary if there exists a map ∂ : Q → {−1, 1} such that φ(i, j) = ∂(i)∂(j)∂(ij), for all i, j ∈ Q.…”