# Barker code

A **Barker code** or **Barker sequence** is a finite sequence of *N* values of +1 and −1,

with the ideal autocorrelation property, such that the off-peak (non-cyclic) autocorrelation coefficients

are as small as possible:

for all .^{[1]}

Only nine^{[2]} Barker sequences are known, all of length *N* at most 13.^{[3]} Barker's 1953 paper asked for sequences with the stronger condition

Only four such sequences are known, shown in bold in the table below.^{[4]} http://paper.uscip.us/jaece/JAECE.2014.1003.pdf

## Known Barker codes

Here is a table of all known Barker codes, where negations and reversals of the codes have been omitted. A Barker code has a maximum autocorrelation sequence which has sidelobes no larger than 1. It is generally accepted that no other perfect binary phase codes exist.^{[5]}^{[6]} (It has been proven that there are no further odd-length codes,^{[7]} nor even-length codes of *N* < 10^{22}.^{[8]})

Length | Codes | Sidelobe level ratio^{[9]}^{[10]} | |
---|---|---|---|

2 | +1 −1 | +1 +1 | −6 dB |

3 | +1 +1 −1 | −9.5 dB | |

4 | +1 +1 −1 +1 | +1 +1 +1 −1 | −12 dB |

5 | +1 +1 +1 −1 +1 | −14 dB | |

7 | +1 +1 +1 −1 −1 +1 −1 | −16.9 dB | |

11 | +1 +1 +1 −1 −1 −1 +1 −1 −1 +1 −1 | −20.8 dB | |

13 | +1 +1 +1 +1 +1 −1 −1 +1 +1 −1 +1 −1 +1 | −22.3 dB |

Barker codes of length *N* equal to 11 and 13 ( A011758 A011759) are used in direct-sequence spread spectrum and pulse compression radar systems because of their low autocorrelation properties (The sidelobe level of amplitude of the Barker codes is 1/N that of the peak signal).^{[11]} A Barker code resembles a discrete version of a continuous chirp, another low-autocorrelation signal used in other pulse compression radars.

The positive and negative amplitudes of the pulses forming the Barker codes imply the use of biphase modulation or binary phase-shift keying; that is, the change of phase in the carrier wave is 180 degrees.

Similar to the Barker codes are the complementary sequences, which cancel sidelobes exactly when summed; the even-length Barker code pairs are also complementary pairs. There is a simple constructive method to create arbitrarily long complementary sequences.

For the case of cyclic autocorrelation, other sequences have the same property of having perfect (and uniform) sidelobes, such as prime-length Legendre sequences and maximum length sequences (MLS). Arbitrarily long cyclic sequences can be constructed.

## Barker modulation

In wireless communications, sequences are usually chosen for their spectral properties and for low cross correlation with other sequences likely to interfere. In the 802.11b standard, an 11-chip Barker sequence is used for the 1 and 2 Mbit/sec rates. The value of the autocorrelation function for the Barker sequence is 0 or −1 at all offsets except zero, where it is +11. This makes for a more uniform spectrum, and better performance in the receivers.^{[12]}

## References

- ↑ Barker, R. H. (1953). "Group Synchronizing of Binary Digital Sequences".
*Communication Theory*. London: Butterworth. pp. 273–287. - ↑ https://oeis.org/A091704
- ↑ Borwein, Peter; Mossinghoff, Michael J. (2008). "Barker sequences and flat polynomials". In James McKee; Chris Smyth.
*Number Theory and Polynomials*. LMS Lecture Notes.**352**. Cambridge University Press. pp. 71–88. ISBN 978-0-521-71467-9. - ↑ Using different pulse shape in Barker code also improves certain Autocorrelation properties.
- ↑ Weisstein, Eric W. "Barker Code".
*MathWorld*. - ↑ http://www.math.wpi.edu/MPI2008/TSC/TSC-MPI.pdf
- ↑ Turyn and Storer, "On binary sequences", Proceedings of the AMS, volume 12 (1961), pages 394–399
- ↑ Leung, K., and Schmidt, B., "The Field descent method", Design, Codes and Cryptography, volume 36, pages 171–188
- ↑ http://www.radartutorial.eu/08.transmitters/Intrapulse%20Modulation.en.html
- ↑ http://www.cacs.louisiana.edu/~library/TR/TR_pdf/TR_2006/TR_darwich_2006-4-1.pdf
- ↑ Introduction to Radar Systems, 3rd Edition, Merrill I. Skolnik, McGraw–Hill, 2001
- ↑ "802.11 & 802.11b" cp.literature.agilent.com