On the Autocorrelation Distributions of Sidel´nikov Sequences

For a prime and positive integers and such that , Sidel\´nikov introduced -ary sequences (called Sidel\´nikov sequences) of period , the out-of-phase autocorrelation magnitude of which is upper bounded by . In this correspondence, we derived the autocorrelation distributions, i.e., the values and the number of occurrences of each value of the autocorrelation function of Sidel\´nikov sequences. The frequency of each autocorrelation value of an -ary Sidel\´nikov sequence is expressed in terms of the cyclotomic numbers of order . It is also pointed out that the total number of distinct autocorrelation values is dependent not only on but also on the period of the sequence, but always less than or equal to .