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) coefficients

are as small as possible:

for all ${\displaystyle 1\leq v<N}$.

Only nine Barker sequences are known, all of length N at most 13. Barker's 1953 paper asked for sequences with the stronger condition

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

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. (It has been proven that there are no further odd-length codes, nor even-length codes of N < 10²².)

Barker codes of length N equal to 11 and 13 (   ) 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). A Barker code resembles a discrete version of a continuous chirp, another low-autocorrelation signal used in other pulse compression radars.

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