  We show that if a Barker sequence of length $n>13$ exists, then either
$n=3979201339721749133016171583224100$, or $n > 4\cdot10^{33}$. This improves
the lower bound on the length of a long Barker sequence by a factor of nearly
2000. We also obtain 18 additional integers $n<10^{50}$ that cannot be ruled
out as the length of a Barker sequence, and find more than 237000 additional
candidates $n<10^{100}$. These results are obtained by completing extensive
searches for Wieferich prime pairs and using them, together with a number of
arithmetic restrictions on $n$, to construct qualifying integers below a given
bound. We also report on some updated computations regarding open cases of the
circulant Hadamard matrix problem.
