Barker code

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Graphical representation of a Barker-7 code
Graphical representation of a Barker-7 code
Autocorrelation function of a Barker-7 code
Autocorrelation function of a Barker-7 code

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

aj for j = 1, \cdot\cdot\cdot, N

such that

|\sum_{j=1}^{N-v} a_j a_{j+v}| \le 1\,

for all 1 \le v < N.

Here is a table of all known optimal Barker codes, where negations and reversals of the codes have been omitted. Optimal is defined as having a maximum autocorrelation of 1 (when codes are not aligned).

Known Barker Codes
Length Codes
2 +1 −1 +1 +1
3 +1 +1 −1
4 +1 −1 +1 +1 +1 −1 −1 −1
5 +1 +1 +1 −1 +1
7 +1 +1 +1 −1 −1 +1 −1
11 +1 +1 +1 −1 −1 −1 +1 −1 −1 +1 −1
13 +1 +1 +1 +1 +1 −1 −1 +1 +1 −1 +1 −1 +1

Barker codes of length 11 and 13 are used in direct-sequence spread spectrum and Pulse Compression Radar systems because of their low autocorrelation properties.

Barker codes utilize biphase modulation; that is, the change of phase in the carrier wave is 180 degrees.

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