Conjectures about Hadamard matrices

The purpose of this paper is to discuss some conjectures related to the construction of Hadamard matrices. We review the results of a computer experiment concerning the extendability of row-Hadamard matrices, generating new conjectures and giving a counterexample to one suggested by Vijayan. An earlier computer investigation by Hall is discussed. We also connect the standard conjectures on the existence of real and generalized Hadamard matrices to the ramification of primes in cyclotomic extensions.