On the asymptotic existence of cocyclic Hadamard matrices

Let q be an odd natural number. We prove there is a cocyclic Hadamard matrix of order 2 10 + t q whenever t ⩾ 8 ⌊ log 2 ( q − 1 ) 10 ⌋ . We also show that if the binary expansion of q contains N ones, then there is a cocyclic Hadamard matrix of order 2 4 N − 2 q .