Barker codes have a maximum autocorrelation sequence, which has off-peak
autocorrelations no larger than 1.

A *correlation sidelobe* is the correlation of a
codeword with a time-shifted version of itself. The correlation sidelobe,
*C*_{k}, for a
*k*-symbol shift of an *N*-bit code
sequence, {*X*_{j}}, is

For *j*=1, 2, 3,..., *N*,
*X*_{j} is an individual code symbol
that is equal to +1 or –1. The adjacent symbols are assumed to be 0.

The output code is in a bipolar format with 0 and 1 mapped to 1 and –1. The maximum known
Barker code length is 13. The short length and low correlation sidelobes make Barker
codes useful for frame synchronization in digital communications systems. The Barker
code generator outputs the Barker codes listed in this table.

Barker Code Length | Barker Code | Sidelobe Level |
---|

`1`
| [–1] | 0 dB |

`2`
| [–1; 1] | –6 dB |

`3`
| [–1; –1; 1] | –9.5 dB |

`4`
| [–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 |