In this paper we construct new classes of optimal frequency-hopping sequences (FHSs) with respect to the Lempel-Greenberger bound and the Peng-Fan bound by interleaving techniques which are used to construct a sequence of length kN from k sequences of length N. We first give two generic constructions for optimal FHS sets from some known optimal FHS sets by interleaving techniques and present some examples of new optimal FHS sets. We then design new optimal FHSs whose parameters include those of the known optimal constructions with length kN and frequency set size n for some positive integers k and N. We also construct optimal FHS sets of length kp from power-residue sequences for any odd prime p and 2 les k < p. In particular, our constructions give several new parameters not covered in the literature, which are summarized in I and II.

It is well known that the magnitude of the cross correlation between any distinct constant multiple sequences of anM-ary power residue sequence of periodpis upper bounded byradicp+2 and that of anM-ary Sidel'nikov sequence of periodpm-1 is upper bounded byradic{pm} +3, wherepis a prime andmis a positive integer. In this paper, we first show that their cross-correlation functions are closely related to Jacobi sums and cyclotomic numbers. We then derive the cross-correlation distribution of constant multiple sequences of anM-ary power residue sequence. In the case of constant multiple sequences of anM-ary Sidel'nikov sequence, we get the possible cross-correlation values whose occurrence numbers are expressed in terms of the cyclotomic numbers of orderMand are possibly zero.

A(v,l, lambda)-FHS denotes a frequency-hopping sequence of lengthvover a frequency set of sizelwith maximum out-of-phase Hamming autocorrelationlambda. Recently, Ding and Yin constructed two FHS families for a prime powerqsatisfyingq=ef+1 with positive integerseandf. Theorems 4 and 5 in their paper claim that these two FHS families include optimal(q-1,e,f)-FHSs and(q-1,e+1,f-1)-FHSs with respect to the Lempel-Greenberger bound, respectively. In this paper, we give counterexamples and make corrections to them. Furthermore, we observe that these FHSs are closely related to Sidel'nikov sequences. Based on our results on the spectrum of their Hamming autocorrelation values, we also correct the theorem on the spectrum of Hamming distances of nearly equidistant codes derived by Sidel'nikov.

In this paper we construct near-optimal frequency-hopping sequences whose maximum Hamming autocorrelation is given by λopt+ 1 where λoptis the optimal Hamming autocorrelation value with respect to the Lempel-Greenberger bound. Their period, frequency set size, and maximum Hamming autocorrelation are given as pq, m and pq−1/m + 1, respectively, where p and q are distinct odd primes, and m is an even common divisor of p − 1 and q − 1.

In this paper we construct M-ary sequence families from power residue sequences of odd prime period p and their constant multiple sequences using the shift-and-add method, when M is a divisor of p - 1. We show that the maximum correlation values of the proposed sequence families of size p + M - 2 and (M - 1)2(p - 1)/2 + M - 1 are upper bounded by 2radic(p) + 5 and 3radic(p) + 4, respectively.

In this correspondence, we study the statistical stability properties of pm-periodic binary sequences in terms of their linear complexity and k-error linear complexity, where p is n prime number and 2 is a primitive root modulo p2. We show that their linear complexity and k-error linear complexity take a value only from some specific ranges. We then present the minimum value k for which the k-error linear complexity is strictly less than the linear complexity in a new viewpoint different from the approach by Meidl. We also derive the distribution of pm-periodic binary sequences with specific k-error linear complexity. Finally, we get an explicit formula for the expectation value of the k-error linear complexity and give its lower and upper bounds, when k les [p/2].