Balanced quadriphase sequences with optimal periodic correlation properties constructed by real-valued bent functions

The real-valued bent function was previously introduced by the authors (1991) as a generalization of the usual p-ary bent function, p a prime, in such a way that the range of the function is the set of real numbers, i.e. not restricted to GF(p). The real-valued bent function was used to construct a family of 2^n/2 balanced quadriphase sequences of period 2ⁿ-1 with optimal periodic correlation properties, where n is a multiple of four. A class of real-valued bent functions that map the set of all the m-tuples over GF(2) into the set {0,1/2,1,3/2} for an arbitrary m is given. This is applied to generalize a previous construction to the case where n is even, i.e. not restricted to a multiple of four. It is also shown that the quadriphase sequences given by T. Novosad can be considered as one kind of sequence constructed by real-valued bent functions. Conditions are given for some families of the quadriphase sequences constructed by some real-valued bent functions to be balanced. The exact distributions of the periodic correlation values are derived for the families of the balanced quadriphase sequences