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.

New Classes of Optimal Frequency-Hopping Sequences by Interleaving Techniques

In this paper, a new generalized cyclotomy with respect to a positive odd integer is introduced, and a construction of frequency hopping sequence sets and two constructions of frequency hopping sequences are proposed as its applications. The frequency hopping sequence sets and frequency hopping sequences obtained in this paper can be optimal with respect to the Peng-Fan bound and Lempel-Greenberger bound, respectively. Further, the length of sequences in the optimal frequency hopping sequence sets can be any odd integer larger than 3. Some of them have new parameters.

Optimal Frequency Hopping Sequences of Odd Length

In this paper, we aim at designing sets of binary sequences with good aperiodic/periodic auto- and cross-correlation functions for multiple-input multiple-output (MIMO) radar systems. We show that such a set of sequences can be obtained by minimizing a weighted sum of peak sidelobe level (PSL) and integrated sidelobe level (ISL) with the binary element constraint at the design stage. The sets of designed sequences are neighboring the lower bound on ISL and have a better PSL than the best-known structured sets of binary sequences. To formulate the problem, we introduce a Pareto-objective of weighted auto- and cross-correlation functions by establishing a multi-objective NP-hard constrained optimization problem. Then, by using the block coordinate descent framework, we propose an efficient monotonic algorithm based on fast Fourier transform, to minimize the multi-dimensional objective function. Numerical results illustrate the superior performance of the proposed algorithm in comparison with the state-of-the-art methods.

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Designing Sets of Binary Sequences for MIMO Radar Systems

A frequency-hopping sequence (FHS) of length v and frequency set size M is called a (v,M,?)-FHS if its maximum out-of-phase Hamming autocorrelation is ?. Three new classes of optimal FHSs with respect to the Lempel-Greenberger bound are presented in this paper. First, new optimal (p,M,f)-FHSs are constructed when p = Mf +1 is an odd prime such that f is even and p ? 3 mod 4 . And then, a construction for optimal (kp,p,k)-FHSs is given for any odd prime p and a positive integer K < p such that k = 2,4,p1,p1(p1 + 2 ),2m -1,or p1 m-1, where p1 and p1+2 are odd primes. Finally, several new optimal FHSs with maximum out-of-phase Hamming autocorrelation 1 or 2 are also presented. In particular, the existence of optimal (v,N,1)-FHSs is proven for any integer N ? 3 and any integer v with N +1 ? v ? 2 N-1, as well as the existence of optimal (2N +1,N,2)-FHSs is shown for any integer N ? 3. These classes of optimal FHSs have new parameters which are not covered in the literature.

Optimal Frequency-Hopping Sequences With New Parameters

In this paper, two generic constructions of optimal frequency-hopping sequence (FHS) sets employing d-form functions with difference-balanced property are presented. They generalize the previous constructions of optimal FHS sets using m-sequences and produce new optimal FHS sets that cannot be produced by the earlier constructions. By choosing appropriate d-form functions with difference-balanced property, both constructions lead to FHSs with large linear complexity. In addition, one of the proposed constructions gives new optimal parameters of FHS sets.

New Constructions for Optimal Sets of Frequency-Hopping Sequences

A (v,l, lambda)-FHS denotes a frequency-hopping sequence of length v over a frequency set of size l with maximum out-of-phase Hamming autocorrelation lambda. Recently, Ding and Yin constructed two FHS families for a prime power q satisfying q=ef+1 with positive integers e and f . 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.

On the Sidel'nikov Sequences as Frequency-Hopping Sequences

In this paper, we construct four M-ary sequence families from a power residue sequence of odd prime period p and its 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 are upper-bounded by 2radicp +5 or 3radicp +4. In addition, we prove that the linear complexity of each sequence in the proposed families is either p-1 or p-[(p-1)/(M)]-1 . We also construct an M-ary sequence family from Sidel'nikov sequences of period p m-1 by applying the same method, when M is a divisor of p m-1. The proposed sequence family F tilde s has larger size than the known M-ary Sidel'nikov sequence families, whereas they all have the same upper bound on the maximum correlation.

New $M$ -Ary Sequence Families With Low Correlation and Large Size

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].

On the $k$ -Error Linear Complexity of $p^{m}$ -Periodic Binary Sequences

In this paper we present a construction method for quaternary sequences from a binary sequence of even period, which preserves the period and autocorrelation of the given binary sequence. By applying the method to the binary sequences with three-valued autocorrelation, we construct new quaternary sequences with three-valued autocorrelation, which are balanced or almost balanced. In particular, we construct new balanced quaternary sequences whose autocorrelations are three-valued and have out-of-phase magnitude 2, when their periods are N = pm - 1 and N ≡ 2 (mod 4) for any odd prime p and any odd integer m. Their out-of-phase autocorrelation magnitude is the known optimal value for N ≠ 2, 4, 8, and 16.

Yun Kyoung Han

Papers

New Classes of Optimal Frequency-Hopping Sequences by Interleaving Techniques

On the Sidel'nikov Sequences as Frequency-Hopping Sequences

New $M$ -Ary Sequence Families With Low Correlation and Large Size

On the $k$ -Error Linear Complexity of $p^{m}$ -Periodic Binary Sequences

New Quaternary Sequences with Even Period and Three-Valued Autocorrelation