# constructing “pseudonoise” sequences other than (2^n)-1? (low cyclical autocorrelation)

Pseudonoise LFSR sequences of length $N = 2^k-1$ have the nice property that their cyclical autocorrelation is $N$ when the sequence is lined up with itself, and $-1$ elsewhere.

Is there a way to construct sequences of other lengths, that their cyclical correlation is close to $0$ or $-1$ when not lined up? If not, why not?

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hmm, looks like there's a lot of research on this: signalslab.marstu.net/?page_id=1769 –  Jason S Apr 5 '12 at 18:52
Look for Legendre sequences among others. –  Dilip Sarwate Apr 5 '12 at 21:19
Are LFSR sequences sequences of $1$s and $-1$s? If so, then you can construct a sequence with the desired autocorrelation properties using the quadratic residues modulo a prime congruent to 3 (mod 4). For example, if $p=19$, then the quadratic residues (perfect squares) in the field are 0, 1, 4, 5, 6, 7, 9, 11, 16, 17. The sequence with $1$ in each of the listed positions and $-1$ in every other position has periodic autocorrelation $-1$ when the sequences are not lined up and autocorrelation 19 when they are lined up.