A Note on the Circulant Hadamard Conjecture
نویسنده
چکیده
This note reports work in progress in connection with Ryser’s conjecture on circulant Hadamard matrices.
منابع مشابه
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.
متن کاملNew restrictions on possible orders of circulant Hadamard matrices
We obtain several new number theoretic results which improve the field descent method. We use these results to rule out many of the known open cases of the circulant Hadamard matrix conjecture. In particular, the only known open case of the Barker sequence conjecture is settled.
متن کاملNonexistence of Abelian Difference Sets: Lander’s Conjecture for Prime Power Orders
In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order > 4 and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander’s conjecture which asserts that there is no abelian group with a cyclic Sylow p-subgroup containing a difference set of order divisible by p. We verify Lander’s conjecture for all differenc...
متن کاملNew Duality Operator for Complex Circulant Matrices and a Conjecture of Ryser
We associate to any given circulant complex matrix C another one E(C) such that E(E(C)) = C∗ the transpose conjugate of C. All circulant Hadamard matrices of order 4 satisfy a condition C4 on their eigenvalues, namely, the absolute value of the sum of all eigenvalues is bounded above by 4. We prove by a “descent” that uses our operator E that the only circulant Hadamard matrices of order n > 4,...
متن کاملAnalytic Aspects of the Circulant Hadamard Conjecture
We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for |q0| = . . . = |qN−1| = 1 the quantity Φ = ∑ i+k=j+l qiqk qjql satisfies Φ ≥ N, with equality if and only if q = (qi) is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems,...
متن کامل