a1 Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada email [email protected]
a2 Department of Mathematics, Davidson College, Davidson, NC 28035-6996, USA email [email protected]
We show that if a Barker sequence of lengthexists, then either n 3 979 201 339 721 749 133 016 171 583 224 100, or . This improves the lower bound on the length of a long Barker sequence by a factor of nearly . We also obtain eighteen additional integers that cannot be ruled out as the length of a Barker sequence, and find more than 237 000 additional candidates . These results are obtained by completing extensive searches for Wieferich prime pairs and using them, together with a number of arithmetic restrictions on , to construct qualifying integers below a given bound. We also report on some updated computations regarding open cases of the circulant Hadamard matrix problem.
(Received July 02 2013)
(Online publication April 2014)
2010 Mathematics Subject Classification