New Self-Dual Codes from Cocyclic Hadamard Matrices

The structure of cocyclic Hadamard matrices allows for a much faster and more systematic search for binary, self-dual codes. Here, we consider \(\mathbf{Z}_{2}^{2} \times \mathbf{Z}_{t}\)-cocyclic Hadamard matrices for \(t = 3, 5, 7,\) and \(9\) to yield binary
self-dual codes of lengths \(24, 40, 56,\) and \(72\). We show that the extended Golay code cannot be obtained as a member of this class and also demonstrate the existence of four apparently new codes – a \([56, 28, 8]\) code and three \([72, 36, 8]\) codes.