Submatrices of Hadamard matrices: complementation results

Submatrices of Hadamard matrices: complementation results

Two submatrices A,D of a Hadamard matrix H are called complementary if, up to a permutation of rows and columns, H = [

Nonexistence Results for Hadamard-like Matrices

The class of square (0, 1,−1)-matrices whose rows are nonzero and mutually orthogonal is studied. This class generalizes the classes of Hadamard and Weighing matrices. We prove that if there exists an n by n (0, 1,−1)-matrix whose rows are nonzero, mutually orthogonal and whose first row has no zeros, then n is not of the form pk, 2pk or 3p where p is an odd prime, and k is a positive integer.

Reconstruction of matrices from submatrices

For an arbitrary matrix A of n×n symbols, consider its submatrices of size k×k, obtained by deleting n−k rows and n−k columns. Optionally, the deleted rows and columns can be selected symmetrically or independently. We consider the problem of whether these multisets determine matrix A. Following the ideas of Krasikov and Roditty in the reconstruction of sequences from subsequences, we replace t...

Jacket matrices constructed from Hadamard matrices and generalized Hadamard matrices

The number of submatrices of a given type in a Hadamard matrix and related results

Let a, b be fixed positive integers. It is proved that if every relatively large sub-matrix of a i-l-matrix N contains about the same number of both entries, then all 2 " b a by b matrices occur asymptotically the same number of times as a submatrix of N. A similar statement is shown to hold for graphs with uniformly distributed edges.