A Walsh-Fourier approach to the circulant Hadamard conjecture

We describe an approach to the circulant Hadamard conjecture based on Walsh-Fourier analysis. We show that the existence of a circulant Hadamard matrix of order $n$ is equivalent to the existence of a non-trivial solution of a certain homogenous linear system of equations. Based on this system, a possible way of proving the conjecture is proposed.Comment: 8 page

Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulse compression, and surface wave encodings

AbstractIf a Williamson matrix of order 4w exists and a special type of design, a set of Baumert-Hall units of order 4t, exists, then there exists a Hadamard matrix of order 4tw. A number of special Baumert-Hall sets of units, including an infinite class, are constructed here; these give the densest known classes of Hadamard matrices. The constructions relate to various topics such as pulse compression and image encodings

New Constructions of Complete Non-cyclic Hadamard Matrices, Related Function Families and LCZ Sequences

Abstract. A Hadamard matrix is said to be completely non-cyclic (CNC) if there are no two rows (or two columns) that are shift equivalent in its reduced form. In this paper, we present three new constructions of CNC Hadamard matrices. We give a primary construction using a flipping operation on the submatrices of the reduced form of a Hadamard matrix. We show that, up to some restrictions, the Kronecker product preserves the CNC property of Hadamard matrices and use this fact to give two secondary constructions of Hadamard matrices. The applications to construct low correlation zone sequences are provided

The field descent method

10.1007/s10623-004-1703-7Designs, Codes, and Cryptography362171-188DCCR