A Walsh-Fourier approach to the circulant Hadamard conjecture (original) (raw)

Cryptography and Communications, 2010

In this note we use combinatorial methods to show that the unique, up to equivalence, 5 × 5 (1, −1)-matrix with determinant 48, the unique, up to equivalence, 6 × 6 (1,−1)-matrix with determinant 160, and the unique, up to equivalence, 7 × 7 (1,−1)-matrix with determinant 576, all cannot be embedded in the Hadamard matrix of order 8. We also review some properties of Sylvester Hadamard matrices, their Smith Normal Forms, and pivot patterns of Hadamard matrices when Gaussian Elimination with complete pivoting is applied on them. The pivot values which appear reconfirm the above non-embedding results.

Invitation to Hadamard matrices

An Hadamard matrix is a square matrix HinMN(pm1)H\in M_N(\pm1)HinMN(pm1) whose rows and pairwise orthogonal. More generally, we can talk about the complex Hadamard matrices, which are the square matrices HinMN(mathbbC)H\in M_N(\mathbb C)HinMN(mathbbC) whose entries are on the unit circle, ∣Hij∣=1|H_{ij}|=1∣Hij∣=1, and whose rows and pairwise orthogonal. The main examples are the Fourier matrices, FN=(wij)F_N=(w^{ij})FN=(wij) with w=e2pii/Nw=e^{2\pi i/N}w=e2pii/N, and at the level of the general theory, the complex Hadamard matrices can be thought of as being some sort of exotic, generalized Fourier matrices. We discuss here the basic theory of the Hadamard matrices, real and complex, with emphasis on the complex matrices, and their geometric and analytic aspects.

SOME NEW EXAMPLES OF CIRCULANT PARTIAL HADAMARD MATRICES OF TYPE

Advances and Applications in Mathematical Sciences, 2022

Using general possible entries and sign pattern we have forwarded here new examples of Circulant Partial Hadamard matrices of the types   12 6 4   H and  . 28 14 4   H This paper contains basic information about Hadamard matrices, partial Hadamard matrices and circulant partial Hadamard matrices. The recent status of Hadamard matrix conjecture is also included. A table of number of inequivalent circulant partial Hadamard matrices of the type

Sufficient conditions for a conjecture of Ryser about Hadamard Circulant matrices

Linear Algebra and its Applications, 2012

Let H be a Hadamard Circulant matrix of order n = 4h 2 where h > 1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial n (t) are bounded by a constant independent of n. Then for all the ϕ(n) n-th primitive roots w of 1, P(w) √ n is not an algebraic integer in the cyclotomic field K = Q(w), where P(t) is the representer polynomial of H and ϕ is the Euler function. This implies that P(w) is not a real number.

Hadamard ideals and Hadamard matrices with two circulant cores

European Journal of Combinatorics, 2006

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, to systematize the application of Computational Algebra methods for this construction. We use the Hadamard ideal formalism to perform exhaustive search constructions of Hadamard matrices with two circulant cores for the twelve orders 20, 24, 28, 32, 36, 40, 44, 48, 52. The total number of such Hadamard matrices is proportional to the square of the parameter. We use the Hadamard ideal formalism to compute the proportionality constants for the twelve orders listed above. Finally, we use the Hadamard ideal formalism to improve the lower bounds for the number of inequivalent Hadamard matrices for the seven orders 44, 48, 52, 56, 60, 64, 68.

On the Complete Pivoting Conjecture for Hadamard Matrices of Small Orders

Journal of Research and Practice in Information Technology

In this paper we study explicitly the pivot structure of Hadamard matrices of small orders 16, 20 and 32. An algorithm computing the (n -j) x (n -j) minors of Hadamard matrices is presented and its implementation for n = 12 is described. Analytical tables summarizing the pivot patterns attained are given.

5TH Workshop on Real and Complex Hadamard Matrices

2017

Progressing on a survey of the state of the art of cocyclic constructions for Hadamard matrices, we will prove that the family of Goethals-Seidel Hadamard matrices is (pseudo)-cocyclic over a certain family of quasigroups. Therefore, what we call (pseudo)-cocyclic (structured) Hadamard matrices over (quasi)groups, precisely those matrices satisfying some kind of cocycle Hadamard test known so far, include most of the well-known and proli c constructions of Hadamard matrices. We will discuss also di erent approaches in order to look for large matrices of this type. This is joint work with J.A. Armario, R.M. Falcón, M.D. Frau, M.B. Güemes, F. Gudiel and I. Kotsireas. Generalized binary sequences from quasi-orthogonal cocycles José A. Armario FRI Universidad de Sevilla, Spain 11:15

On the non-existence of generalized Hadamard matrices

Journal of Statistical Planning and Inference, 2000

It is shown that the solvability of certain quadratic forms is necessary for the existence of some generalized Hadamard matrices. The number-theoretic consequences of this are explored. In particular, non-existence results for generalized "Butson" Hadamard matrices BH(h; p f ) and BH(h; 2p f ) are given, where p is a prime of the form 4k + 3.

On the asymptotic existence of complex Hadamard matrices

Journal of Combinatorial Designs, 1997

Let N = N (q) be the number of nonzero digits in the binary expansion of the odd integer q. A construction method is presented which produces, among other results, a block circulant complex Hadamard matrix of order 2 α q, where α ≥ 2N − 1. This improves a recent result of Craigen regarding the asymptotic existence of Hadamard matrices. We also present a method that gives complex orthogonal designs of order 2 α+ 1 q from complex orthogonal designs of order 2 α. We also demonstrate the existence of a block circulant complex Hadamard matrix of order 2 β q, where β = 4 log 2 (q−1) 10

A Walsh-Fourier approach to the circulant Hadamard conjecture (original) (raw)

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