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Barker Code


A Barker code is a string of digits a_i=+/-1 of length l>=2 such that

 |sum_(i=1)^(l-k)a_ia_(i+k)|<=1

for all 1<=k<l. Barker codes are used for pulse compression of radar signals. There are Barker codes of lengths 2, 3, 4, 5, 7, 11, and 13, and it is conjectured that no longer Barker codes exist. A list of known Barker codes up to reversal of digits and negation is given below.

lengthcode
2+-, ++
3++-
4+-++, +---
5+++-+
7+++--+-
11+++---+--+-
13+++++--++-+-+

The number of candidate codes of length n is therefore equal to the number of n-bead black-white reversible strings 1, 2, 3, 6, 10, 20, 36, 72, ... (OEIS A005418), while the numbers of Barker codes of order l=2, 3, ... are 2, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, and 0 for all higher n (OEIS A091704).


Portions of this entry contributed by David Terr

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References

Barker, R. H. "Group Synchronizing of Binary Digital Sequences." In Communication Theory. London: Butterworth, pp. 273-287, 1953.Lüke, H. D. Korrelationssignale. Berlin: Springer-Verlag, 1992.Sloane, N. J. A. Sequences A05418/M0771 and A091704 in "The On-Line Encyclopedia of Integer Sequences."Stimson, G. W. Introduction to Airborne Radar, 2nd ed. Raleigh, NC: SciTech, p. 172, 1998.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 223, 1995.

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Barker Code

Cite this as:

Terr, David and Weisstein, Eric W. "Barker Code." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BarkerCode.html

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