×

Wieferich pairs and Barker sequences. II. (English) Zbl 1355.11018

Summary: 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.
For Part I, see Des. Codes Cryptography 53, No. 3, 149–163 (2009; Zbl 1228.11032).

MSC:

11B83 Special sequences and polynomials
94A55 Shift register sequences and sequences over finite alphabets in information and communication theory
05-04 Software, source code, etc. for problems pertaining to combinatorics
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
11A07 Congruences; primitive roots; residue systems

Citations:

Zbl 1228.11032
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/s10623-011-9493-1 · Zbl 1242.15027 · doi:10.1007/s10623-011-9493-1
[2] DOI: 10.1007/s10623-004-1703-7 · Zbl 1066.05037 · doi:10.1007/s10623-004-1703-7
[3] DOI: 10.1007/BF00124212 · Zbl 0762.94009 · doi:10.1007/BF00124212
[4] DOI: 10.1016/0097-3165(90)90046-Y · Zbl 0705.94012 · doi:10.1016/0097-3165(90)90046-Y
[5] DOI: 10.1007/s10623-009-9301-3 · Zbl 1228.11032 · doi:10.1007/s10623-009-9301-3
[6] DOI: 10.1090/S0002-9939-1961-0125026-2 · doi:10.1090/S0002-9939-1961-0125026-2
[7] DOI: 10.2140/pjm.1965.15.319 · Zbl 0135.05403 · doi:10.2140/pjm.1965.15.319
[8] DOI: 10.1137/0202017 · Zbl 0274.05106 · doi:10.1137/0202017
[9] DOI: 10.1090/S0894-0347-99-00298-2 · Zbl 0939.05016 · doi:10.1090/S0894-0347-99-00298-2
[10] Eliahou, Enseign. Math. (2) 38 pp 345– (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.