On Permanents of Sylvester Hadamard Matrices

It is well-known that a Sylvester Hadamard matrix can be described by means of a cocycle. In this paper it is shown that the additional internal structure in a Sylvester Hadamard matrix, provided by the cocycle, is su cient to guarantee some kind of reduction in the computational complexity of calculating its permanent using Ryser's formula. A Hadamard matrix H of order n is an n× n matrix with elements ±1 and HHT = nI. A Hadamard matrix is said to be normalized if it has its rst row and column all 1′s. We can always normalize a Hadamard matrix by multiplying rows and columns by −1. It is well-known that n is eihter 2 or a multiple of 4 and it is conjectured that Hadamard matrices exist for every n ≡ 0 mod 4 (see [7]). Sylvester in 1867 noted that given a Hadamard matrix H of order n, then [ H H H −H ] is a Hadamard matrix of order 2n. Matrices of this form are called Sylvester Hadamard and de ned for all powers of 2. Below is given the Sylvester Hadamard matrix of order 2