Cocyclic two-circulant core Hadamard matrices

The two-circulant core (TCC) construction for Hadamard matrices uses two sequences with almost perfect autocorrelation to construct a Hadamard matrix. A research problem of K. Horadam asks whether such matrices are cocyclic. Using techniques from the theory of permutation groups, we prove that the order of a cocyclic TCC matrix coincides with the order of a Hadamard matrix of Paley type, of Sylvester type or certain multiples of these orders. We show that there exist cocyclic TCC Hadamard matrices at all allowable orders ⩽ 1000 with at most one exception. Of the four families of TCC matrices known in the literature, we establish that two are cocyclic, prove that one is not cocyclic, and leave one undecided. The undecided family consists of matrices of 2-power order; we show that these are inequivalent to the Sylvester matrices. As a generalisation of the TCC construction, we introduce quadruple-circulant core (QCC) Hadamard matrices; our results give a complete description of the orders that admit cocyclic QCC Hadamard matrices.