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Wieferich pairs and Barker sequences

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Abstract

We show that if a Barker sequence of length n > 13 exists, then either n = 189 260 468 001 034 441 522 766 781 604, or n > 2 · 1030. This improves the lower bound on the length of a long Barker sequence by a factor of more than 107. We also show that all but fewer than 1600 integers n ≤ 4 · 1026 can be eliminated as the order of a circulant Hadamard matrix. These results are obtained by completing extensive searches for Wieferich prime pairs (q, p), which are defined by the relation \({q^{p-1} \equiv1}\) mod p 2, and analyzing their results in combination with a number of arithmetic restrictions on n.

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Authors and Affiliations

  1. Department of Mathematics, Davidson College, Box 6996, Davidson, NC, 28035-6996, USA

    Michael J. Mossinghoff

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  1. Michael J. Mossinghoff
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Correspondence to Michael J. Mossinghoff.

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Communicated by Jonathan Jedwab.

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Mossinghoff, M.J. Wieferich pairs and Barker sequences. Des. Codes Cryptogr. 53, 149–163 (2009). https://doi.org/10.1007/s10623-009-9301-3

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