Circulant partial Hadamard matrices

Circulant Hadamard Matrices

Note. The determinant of a circulant matrix is an example of a group determinant, where the group is the cyclic group of order n. In 1880 Dedekind suggested generalizing the case of circulants (and more generally group de­ terminants for abelian groups) to arbitrary groups. It was this suggestion that led Frobenius to the creation group of representation theory. See [1] and the references therein.

Double Wieferich Pairs and Circulant Hadamard Matrices

We show that all but 4489 integers n with 4 < n ≤ 4 ·1030 cannot occur as the order of a circulant Hadamard matrix. Our algorithm allows us to search 10000 times farther than prior efforts, while substantially reducing memory requirements. The principal improvement over prior methods involves the incorporation of a separate search for double Wieferich prime pairs {p, q}, which have the property...

Hadamard ideals and Hadamard matrices with two circulant cores

We apply Computational Algebra methods to the construction of Hadamard matrices with two circulant cores, given by Fletcher, Gysin and Seberry. We introduce the concept of Hadamard ideal for this construction to systematize the application of Computational Algebra methods. Our approach yields an exhaustive search construction of Hadamard matrices with two circulant cores for this construction f...

Hadamard ideals and Hadamard matrices with circulant core

Computational Algebra methods have been used successfully in various problems in many fields of Mathematics. Computational Algebra encompasses a set of powerful algorithms for studying ideals in polynomial rings and solving systems of nonlinear polynomial equations efficiently. The theory of Gröbner bases is a cornerstone of Computational Algebra, since it provides us with a constructive way of...

Hadamard matrices constructed by circulant and negacyclic matrices