Heuristic algorithms for Hadamard matrices with two circulant cores

We design heuristic algorithms to construct Hadamard matrices with two circulant cores. This hard combinatorial problem can be formulated in terms of objective functions of several binary variables, so that heuristic methodologies can be used. Our algorithms are based on local and tabu search and they use information on the geometry of the objective function landscapes. In addition, we use the supplementary difference sets formalism to detect when solutions of a special structure exist. Using these algorithms we have computed at least one Hadamard matrix with two circulant cores of the sixteen orders 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116. In particular, the Hadamard matrix with two circulant cores of order 116 is constructed here for the first time, indeed it was accidentally reported as known in an earlier paper.