Hadamard matrices of order ? (8 mod 16) with maximal excess

Christos Koukouvinos
Jennifer Seberry, University of Wollongong

Article comments

Christos Koukouvinos and Jennifer Seberry, Hadamard matrices of order ? (8 mod 16) with maximal excess, Discrete Mathematics, 92, (1991), 173-176, also appeared in Selected Papers in Combinatorics - a Volume Dedicated to R.G. Stanton, Topics in Discrete Mathematics, 2, North Holland, New York, 1992.

Abstract

Kounias and Farmakis, in 'On the excess of Hadamard matrices', Discrete Math. 68 (1988) 59-69, showed that the maximal excess (or sum of the elements) of an Hadamard matrix of order h, o(h) for h = 4m(m -1) is given by

o(4m(m - 1))≤4(m - 1)²(2m + 1).

Kharaghani in 'An infinite class of Hadamard matrices of maximal excess' (to appear) showed this maximal excess can be attained if m is the order of a skew-Hadamard matrix. We give another proof of Kharaghani's result, by generalizing an example of Farmakis and Kounias, 'The excess of Hadamard matrices and optimal designs', Discrete Math. 67 (1987) 165-176, and further show that the maximal excess of the bound is attained if m ≡ 2 (mod 4) is the order of a conference matrix.