# Choosing Codes or sequences with excellent Auto-Correlation properties

The Auto-Correlation function of Walsh-Hadamard codewords does not have a good characteristics.

It can have more than one peak and thus, the Walsh-Hadamard codes do not have the best spreading behavior or correlation property.

The cross-correlation function of the Walsh-Hadamard codewords can also be non-zero for a number of time shifts and unsynchronized users can interfere with each other.

My Questions:

Which other codes or sequences show :

A good auto-correlation attribute?
Good Cross-correlation Property ?

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–  Jason R Dec 11 '13 at 15:22

There is a vast literature on this subject. In particular, deleting one bit (the leading bit in one specific implementation) from each of the Walsh-Hadamard sequences and applying a permutation to the remaining $2^n-1$ bits will result in $2^n$ sequences of length $2^n-1$ that consist of

(i) the all-zeroes sequence

(ii) the $2^n-1$ cyclic shifts of a pseudonoise (PN), or maximal-length linear feedback shift register (LFSR), sequence

A PN sequence has ideal autocorrelation properties, but the cross-correlation between a sequence and its cyclic shift has a peak value that is the same as the autocorrelation peak.

For more than you probably want to know, I refer you to the paper D.V. Sarwate and M.B. Pursley, "Cross-correlation properties of pseudorandom and related sequences," Proc. IEEE, vol.68, pp.593-619, May 1980. It includes a detailed discussion of the Gold sequences mentioned by JasonR as well as the small sets and large sets of Kasami sequences that you will encounter in some applications.

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Thank You Sir ! –  user2888999 Dec 11 '13 at 18:01
Sir I got the paper.Please do not mind me asking that are you the same Dilip Sarwate? –  user2888999 Dec 12 '13 at 15:09

Barker codes are typically used in RADAR applications since they have a high correlation peak and low (sometimes constant) secondary lobes.

http://en.wikipedia.org/wiki/Barker_code

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And what if we need a binary Barker code of length more than 13? By the way, Barker codes do not have constant secondary lobes except at length 4 when the sidelobes are identically 0 and we have what is sometimes called a perfect sequence. –  Dilip Sarwate Dec 11 '13 at 21:28