Wieferich pairs and Barker sequences, II

Abstract

We show that if a Barker sequence of length $n>13$ exists, then either n $=$ 3 979 201 339 721 749 133 016 171 583 224 100, or $n > 4\cdot 10^{33}$ . This improves the lower bound on the length of a long Barker sequence by a factor of nearly $2000$ . We also obtain eighteen additional integers $n<10^{50}$ that cannot be ruled out as the length of a Barker sequence, and find more than 237 000 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.