Applied Numerical Mathematics 60 (2010) 370–377 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum Improving the lower bounds on inequivalent Hadamard matrices through orthogonal designs and meta-programming techniques Christos Koukouvinos, Dimitris E. Simos ∗ Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece article info abstract Article history: Received 6 December 2008 Accepted 9 June 2009 Available online 17 June 2009 MSC: 05B20 05B15 90C90 Keywords: Hadamard matrices Orthogonal designs Sequences Non-periodic autocorrelation function Meta-programming In this paper, we construct inequivalent Hadamard matrices based on several new and old full orthogonal designs, using circulant and symmetric block matrices. Not all orthogonal designs produce inequivalent Hadamard matrices, because the corresponding systems of equations do not possess solutions. In addition, we give some new constructions for orthogonal designs derived from sequences with zero autocorrelation. The orthogonal designs used to construct the inequivalent Hadamard matrices are produced from theoretical and algorithmic constructions.  2009 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction An orthogonal design of order n and type (s 1 , s 2 ,..., s k ) denoted OD(n; s 1 , s 2 ,..., s k ) in the commuting variables x 1 , x 2 ,..., x k , is a square matrix D of order n with entries from the set {0, ±x 1 , ±x 2 ,..., ±x k } satisfying DD T = k  i =1 ( s i x 2 i ) I n , where I n is the identity matrix of order n.A Hadamard matrix of order n is an n × n {1, −1}-matrix satisfying HH T = nI n . It is well known that if n is the order of a Hadamard matrix then n is necessarily 1, 2 or a multiple of 4. A weighing matrix W = W (n, w) is a square matrix with entries 0, ±1 having w non-zero entries per row and column and inner product of distinct rows equal to zero. Therefore W satisfies WW T = wI n . The number w is called the weight of W . Orthogonal designs are used in Combinatorics, Statistics, Coding Theory, Telecommunications and other areas. For more details on orthogonal designs see [5,20] and on Hadamard matrices see [3]. It is well known that the maximum number of variables which may appear in an orthogonal design is given by the Radon’s function ρ (n) which is defined by ρ (n) = 8c + 2 d , when n = 2 a b, b odd, a = 4c + d,0  d < 4 (Geramita and Seberry [5]). An OD(m; a 1 ,..., a k ) will be called full orthogonal design, if a 1 +···+ a k = m. * Corresponding author. E-mail addresses: ckoukouv@math.ntua.gr (C. Koukouvinos), dsimos@math.ntua.gr (D.E. Simos). 0168-9274/$30.00  2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2009.06.002