New quaternary sequences with ideal autocorrelation constructed from Legendre sequences

In this paper, for an odd prime p, new quaternary sequences of even period 2p with ideal autocorrelation property are constructed using the binary Legendre sequences of period p. For the new quaternary sequences, two properties which are considered as the major characteristics of pseudo-random sequences are derived. Firstly, the autocorrelation distribution of the proposed quaternary sequences is derived and it is shown that the autocorrelation values of the proposed quaternary sequences are optimal. For both p ≡ 1 mod 4 and p ≡ 3 mod 4, we can construct optimal quaternary sequences while only for p ≡ 3 mod 4, the binary Legendre sequences can satisfy ideal autocorrelation property. Secondly, the linear complexity of the proposed quaternary sequences is also derived by counting non-zero coefficients of the discrete Fourier transform over the finite field F_q which is the splitting field of x^2p - 1. It is shown that the linear complexity of the quaternary sequences is larger than or equal to p or (3p + 1)/2 for p ≡ 1 mod 4 or p ≡ 3 mod 4, respectively.