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Wieferich pairs and Barker sequences, II

Part of: Communication, information Combinatorics Designs and configurations Elementary number theory

Published online by Cambridge University Press:  01 April 2014

Peter Borwein  and
Michael J. Mossinghoff
Peter Borwein
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada email [email protected]
Michael J. Mossinghoff
Affiliation:
Department of Mathematics, Davidson College, Davidson, NC 28035-6996, USA email [email protected]

Abstract

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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.

MSC classification

Secondary: 05B20: Matrices (incidence, Hadamard, etc.) 94A55: Shift register sequences and sequences over finite alphabets 05-04: Explicit machine computation and programs (not the theory of computation or programming) 05B10: Difference sets (number-theoretic, group-theoretic, etc.) 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 24 - 32
Copyright
© The Author(s) 2014 

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