We consider the problem of finding maximal sets of shift-inequivalent decimations of Sidelnikov–Lempel–Cohn–Eastman (SLCE) sequences (as well as the equivalent problem of determining the multiplier groups of the almost difference sets associated with these sequences). This is an open problem that was originally posed in Cohn et al (IEEE Trans Inf Theory 23:38–42, 1977) and that was mentioned more recently as being open in Akiyama (Acta Arithmetica LXXV.2:97–104, 1996). We derive a numerical necessary condition for a residue to be a multiplier of an SLCE almost difference set. Using our necessary condition, we show that if p is an odd prime and S is an SLCE almost difference set in Zp∗, then the multiplier group of S is trivial. Consequently, for each odd prime p, we obtain a family of ϕ(p- 1 ) shift-inequivalent balanced periodic sequences (where ϕ is the Euler-Totient function) each having period p- 1 and nearly perfect autocorrelation.

Almost difference sets, Autocorrelation, CDMA, Cross-correlation, Difference sets, Feedback shift registers, Gauss sums, Jacobi sums, Sidelnikov–Lempel–Cohn–Eastman sequences, Sidel’nikov sequences
Designs, Codes and Cryptography
School of Mathematics and Statistics

Alaca, S, & Millar, G. (Goldwyn). (2019). Shift-inequivalent decimations of the Sidelnikov–Lempel–Cohn–Eastman sequences. Designs, Codes and Cryptography. doi:10.1007/s10623-019-00697-8