Hadamard matrices of order cube plus one

Abstract

2. Definitions and proof. Throughout this paper 7 and / denote the identity matrix and the matrix with 1 in every position respectively, of the order required by the context. An a, b matrix is one in which each element is either a or b. An Hadamard matrix is a 1, —1 matrix 77 of order h such that HHT = hI. (Necessarily either h = 2 or h is divisible by 4.) It is of type 1 if 77+77r = 27. An Hadamard design A is a 0, 1 matrix of order h — such that AAT = ATA = (/j/4)7+(V4-1)J. (Necessarily^ J = JA = (h/2-\)J.) It is of type 1 if A+AT = J-I. If 77 is an Hadamard matrix it can be multiplied by generalized permutation matrices to bring it into the form