Problems of Information Transmission, Vol. 38, No. 4, 2002, pp. 255–267. Translated from Problemy Peredachi Informatsii, No. 4, 2002, pp. 10–23.
Original Russian Text Copyright
2002 by Gabidulin, Shorin.
INFORMATION THEORY AND CODING THEORY
New Sequences with Zero Autocorrelation
E. M. Gabidulin and V. V. Shorin
Received April 2, 2002
Abstract—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 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 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.
Let x =(x
) be a nonzero complex-valued sequence of length n.
Deﬁnition 1. A sequence is called delta-correlated (has zero autocorrelation) if it is orthogonal
to all of its cyclic shifts:
=0,k=1, 2,...,n− 1. (1)
means complex conjugation.
Let ζ be a primitive nth root of unity, i.e., ζ
Deﬁnition 2. The unitary matrix
,i,j=0, 1,...,n− 1,
is called the matrix of a (direct) nth-order discrete Fourier transform.
Deﬁnition 3. The vector
is called the Fourier image of vector x.
The inverse Fourier transform is deﬁned by the matrix
,i,j=0, 1,...,n− 1.
Theorem 1 . Asequencex is delta-correlated if and only if all components of its Fourier
image are of the same magnitude:
,j=0, 1,...,n− 1.
Theorem 1 yields a classiﬁcation of the whole class of delta-correlated sequences and a method
of constructing them: one takes a vector y with components of the same magnitude and applies
the inverse Fourier transform.
2002 MAIK “Nauka/Interperiodica”