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Barker code : ウィキペディア英語版
Barker code

A Barker code or Barker sequence is a finite sequence of ''N'' values of +1 and −1,
:a_j \text j = 1, 2, \dots, N
with the ideal autocorrelation property, such that the off-peak (non-cyclic) autocorrelation coefficients
:c_v = \sum_^ a_j a_
are as small as possible:
:|c_v| \le 1\,
for all 1 \le v < N.
Only nine〔https://oeis.org/A091704〕 Barker sequences are known, all of length ''N'' at most 13. Barker's 1953 paper asked for sequences with the stronger condition
:c_v \in \
only four such sequences are known, shown in bold in the table below.〔
== 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.〔http://www.math.wpi.edu/MPI2008/TSC/TSC-MPI.pdf〕 (It has been proven that there are no further odd-length codes,〔Turyn and Storer, "On binary sequences", Proceedings of the AMS, volume 12 (1961), pages 394–399〕 nor even-length codes of ''N'' < 1022.〔Leung, K., and Schmidt, B., "The Field descent method", Design, Codes and Cryptography, volume 36, pages 171–188〕)
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).〔Introduction to Radar Systems, 3rd Edition, Merrill I. Skolnik, McGraw–Hill, 2001〕 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 2^n-1 maximum length sequences (MLS). Arbitrarily long cyclic sequences can be constructed.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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