A091704 Number of Barker codes (or Barker sequences) of length n up to reversals and negations.

2, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

2,1

Comments

It is conjectured that there are no Barker codes of length > 13.

If there are any nonzero terms for n > 13, they are either at n = 3979201339721749133016171583224100 or at n larger than 4 * 10^33 (Borwein & Mossinghoff, 2014). - Felix Fröhlich, Feb 08 2017

References

R. H. Barker, Group synchronizing of binary digital sequences, in "Communication Theory", Butterworth, London, 1953, pp. 273-287.

H. D. Lueke, Korrelationssignale, Springer 1992.

Links

Table of n, a(n) for n=2..103.

Peter Borwein and Tamas Erdelyi, A note on Barker polynomials, arXiv:1206.5371 [math.NT], 2012.

Peter Borwein and Michael J. Mossinghoff, Wieferich pairs and Barker sequences, II, LMS Journal of Computation and Mathematics, Vol. 17, No. 1 (2014), 24-32.

Shalom Eliahou, Connaît-on toutes les suites de Barker ?, Images des Mathématiques, CNRS, 2022. In French.

Luis H. Gallardo, Ryser's Conjecture and Stochastic matrices, Univ. Brest (France 2024). See p. 2.

Brooke Logan Ogrodnik and Michael J. Mossinghoff, Double Wieferich pairs and circulant Hadamard matrices, ResearchGate, 2015.

Michael J. Mossinghoff, Wieferich pairs and Barker sequences, Designs, Codes and Cryptography, Vol. 53, No. 3 (2009), 149-163.

Kai-Uwe Schmidt and Jürgen Willms, Barker sequences of odd length, Des. Codes Cryptogr. Vol. 80, No. 2 (2016), 409-414.

R. Turyn and J. Storer, On binary sequences, Proceedings of the American Mathematical Society, vol. 12, no. 3, pp. 394-399, 1961.

Jürgen Willms, Counterexamples to Theorem 1 of Turyn's and Storer's paper "On Binary Sequences", arXiv:1404.4833 [math.NT], 2014.

Eric Weisstein's World of Mathematics, Barker Code

Wikipedia, Barker code.

Example

{{+, +},  {+, -}},
{{+, +, -}},
{{+, +, +, -}, {+, +, -, +}},
{{+, +, +, -, +}},
{{+, +, +, -, -, +, -}},
{{+, +, +, -, -, -, +, -, -, +, -}},
{{+, +, +, +, +, -, -, +, +, -, +, -, +}}

Crossrefs

Cf. A011758, A011759, A276690.

Sequence in context: A082451 A121362 A234694 * A175799 A123739 A165575

Adjacent sequences: A091701 A091702 A091703 * A091705 A091706 A091707

Keyword

nonn

Author

Eric W. Weisstein, Jan 30 2004

Status

approved