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.
I believe that such sequences can also be constructed when the length is a product of twin primes. If this is the kind of thing you have in mind, I can try to dig up some references.