The cocyclic Hadamard matrices of order less than 40

Abstract

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.

Classifying Cocyclic Butson Hadamard Matrices

“…By Remark 5.6 and Corollary 5.8, for (n, p) = (10, 5) or p = 3 and n ∈ {6, 24, 30}, there are no cocyclic BH(n, p) at all (so Butson's construction [4] is not cocyclic). Furthermore, a cocyclic BH (12,3), BH (21,3), BH (20,5), or BH(14, 7) is equivalent to a group-developed matrix.…”

“…The search for a relative difference set with parameters (14, 7, 14, 2) ran in under an hour; the test for an RDS (20,5,20,4) took about a day, with most of the time being spent on C 100 . We note additionally that there are theoretical obstructions to the existence of an RDS (21,3,21,7): the system of diophantine signature equations that such a difference set must satisfy does not admit a solution [24].…”

“…Two complementary methods were used for the first step: checking shift orbits for orthogonal cocycles, and constructing relative difference sets. See Subsections 4.2 and 4.3; also, we refer to [21,Section 6], which discusses a classification of cocyclic Hadamard matrices via central relative difference sets. The algorithm for constructing the difference sets in this paper is identical to the one there, and was likewise carried out using M. Röder's GAP package RDS [23].…”

“…New non-existence results are also obtained. We extend MAGMA [1] and GAP [13] procedures implemented previously for 2-cohomology and relative difference sets [12,21,23] to determine the matrices and sort them into equivalence classes.…”

“…It has turned out to be a powerful tool in the study of real Hadamard matrices (see [21] for the most comprehensive classification). A basic strategy, which we follow here, is to use algebraic and cohomological techniques in systematically constructing the designs.…”