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 \(\leqslant 1000\) 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.

Hadamard矩阵的双循环核（TCC）构造使用几乎完全自相关的两个序列来构造Hadamard矩阵。霍拉丹姆（K. Horadam）的研究问题询问这样的矩阵是否是同周期的。使用置换组理论的技术，我们证明了同环TCC矩阵的阶与Paley型，Sylvester型或这些阶的某些倍数的Hadamard矩阵的阶一致。我们表明，在所有允许的阶次\（\ leqslant 1000 \）上都存在同环TCC Hadamard矩阵最多有一个例外。在文献中已知的TCC矩阵的四个家族中，我们确定两个是同循环的，证明一个不是同循环的，还有一个不确定。未定的族由2次幂阶矩阵组成。我们证明它们与Sylvester矩阵不等价。作为TCC构造的概括，我们引入四重循环核（QCC）Hadamard矩阵；我们的结果给出了接纳同周期QCC Hadamard矩阵的阶数的完整描述。