New Sequences with Zero Autocorrelation

New families of unimodular sequences of length $p=3f+1$ with zero autocorrelation are described, $p$ being a prime. The construction is based on employing Gauss periods. It is shown that in this case elements of the sequences are algebraic numbers defined by irreducible polynomials over $\mathbb Z$ of degree 12 (for the first family) and 6 (for the second family). In turn, these polynomials are factorized in some extension of the field $\mathbb Q$ into polynomials of degree, respectively, 4 and 2, which are written explicitly. For $p=13$, using the exhaustive search method, full classification of unimodular sequences with zero autocorrelation is given.