New Self-Dual Codes from Cocyclic Hadamard Matrices

Abstract

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.