Optimal Frequency-hopping Sequences Research Articles

Constructions of optimal low hit zone frequency hopping sequence sets with large family size

<p style='text-indent:20px;'>Frequency hopping sequences with low hit zone is significant for application in quasi synchronous multiple-access systems. In this paper, we obtained two constructions of optimal frequency hopping sequence sets with low hit zone based on interleaving techniques. The presented low hit zone frequency hopping sequence sets are with new and flexible parameters and large family size which can meet the needs of the practical applications. Moreover, all the sequences in the proposed sets are cyclically inequivalent. Some low hit zone frequency hopping sequence sets constructed in literatures are included in our family. The proposed frequency hopping sequence sets with low hit zone are contributed for quasi-synchronous frequency hopping multiple access system to reduce or eliminate multiple-access interference.</p>

Construction of three classes of strictly optimal frequency-hopping sequence sets

<p style='text-indent:20px;'>In this paper, we construct three classes of strictly optimal frequency-hopping sequence (FHS) sets with respect to partial Hamming correlation and family size. The first and second classes are based on the trace map, the third class is based on a generic construction.</p>

Optimal and near-optimal frequency-hopping sequences based on Gaussian period

<abstract><p>Frequency-hopping sequences (FHSs) have a decisive influence on the whole frequency-hopping communication system. The Hamming correlation function plays an important role in evaluating the performance of FHSs. Constructing FHS sets that meet the theoretical bounds is crucial for the research and development of frequency-hopping communication systems. In this paper, three new classes of optimal FHSs based on trace functions are constructed. Two of them are optimal FHSs and the corresponding periodic Hamming autocorrelation value is calculated by using the known Gaussian period. It is shown that the new FHSs are optimal according to the Lempel-Greenberger bound. The third class of FHSs is the near-optimal FHSs.</p></abstract>

Optimal Constructions of Low-Hit-Zone Frequency-Hopping Sequence Sets via Cyclotomy

In complex environments of wireless communication, to improve the communication anti-interference performance, quasi-synchronous frequency-hopping (FH) communication system requires low-hit-zone (LHZ) FH sequence sets with optimal Hamming correlation (HC) properties and flexible parameters. In this paper, based upon cyclotomy theory, two classes of LHZ FH sequence sets with flexible parameters not included in the literatures are designed, and the periodic HC properties of which are analyzed. It turns out that the designed LHZ FH sequence sets are optimal on the maximum periodic HC. And thus the new constructions can be used to provide optimal FH sequence sets for quasi-synchronous FH communication system.

New constructions of optimal frequency hopping sequences

New constructions of optimal frequency hopping sequences

Optimal frequency-hopping sequences based on the decimated m-sequences

Optimal frequency-hopping sequences based on the decimated m-sequences

New classes of strictly optimal low hit zone frequency hopping sequence sets

<p style='text-indent:20px;'>Low hit zone frequency hopping sequences (LHZ FHSs) with favorable partial Hamming correlation properties are desirable in quasi-synchronous frequency hopping multiple-access systems. An LHZ FHS set is considered to be strictly optimal when it has optimal partial Hamming correlation for all correlation windows. In this study, an interleaved construction of new sets of strictly optimal LHZ FHSs is proposed. Strictly optimal LHZ FHS sets with new and flexible parameters are obtained by selecting suitable known optimal FHSs and appropriate shift sequences.

New Extension Constructions of Optimal Frequency-Hopping Sequence Sets

In this paper, a general framework of constructing the optimal frequency-hopping sequence (FHS) sets is presented based on the designated direct product. Under this framework, we obtain infinitely many new optimal FHS sets by combining some one-coincide (OC) sequence sets that are newly constructed in this paper with some known optimal FHS sets. Our constructions are also based on extension method. However, our constructions give new and flexible parameters due to the free choice of OC sequence set. As a result, our constructions allow a great flexibility of choosing parameters of FHS sets for a given frequency-hopping spread spectrum system.

A New Method to Construct Strictly Optimal Frequency Hopping Sequences With New Parameters

In order to evaluate the goodness of frequency hopping (FH) sequence design, the periodic Hamming correlation function is used as an important measure. Usually, the length of correlation window is shorter than the period of the chosen FH sequence, so the study of the partial Hamming correlation of FH sequence is particularly important. If an FH sequence or an FH sequence set has an optimal partial Hamming correlation with respect to the partial Hamming correlation bound for all length of correlation window, then the FH sequence or the FH sequence set is said to be strictly optimal. In this paper, we first prove that there is no strictly optimal FH sequence set of family size $M$ and sequence length $N$ over a frequency slot set of size $q$ with respect to the partial Hamming correlation bound derived by Niu et al. when $N>{q^{2}}/{M}$ and $q\geq 2$ , and that by Cai et al. when $N>{q^{2}}/{M}$ and $q\geq {2N}/({N-2})$ . Furthermore, we define a special partition-type difference packing (DP) called $[N,\nabla,H_{a}^{l}]$ PDP and give several classes of $[N,\nabla,H_{a}^{l}]$ PDPs. Then, we present a new construction of strictly optimal FH sequences. By choosing different PDPs, the FH sequences constructed can give new and flexible parameters. By utilizing this construction method recursively, we can obtain new $[N,\nabla,H_{a}^{l}]$ PDPs, which lead to infinitely many classes of strictly optimal FH sequences with new parameters. Moreover, based upon an $[N,\nabla,H_{a}^{l}]$ PDP, we present a construction of strictly optimal FH sequence sets. By preceding construction method and recursive construction, we can also obtain infinite classes of strictly optimal FH sequence sets which can give new and flexible parameters.

More cyclotomic constructions of optimal frequency-hopping sequences

In this paper, some general properties of the Zeng-Cai-Tang-Yang cyclotomy are studied. As its applications, two constructions of frequency-hopping sequences (FHSs) and two constructions of FHS sets are presented, where the length of sequences can be any odd integer larger than 3. The FHSs and FHS sets generated by our construction are (near-) optimal with respect to the Lempel–Greenberger bound and Peng–Fan bound, respectively. By choosing appropriate indexes and index sets, a lot of (near-) optimal FHSs and FHS sets can be obtained by our construction. Furthermore, some of them have new parameters which are not covered in the literature.

New Extension Interleaved Constructions of Optimal Frequency Hopping Sequence Sets With Low Hit Zone

In this paper, a designated interleaved structure of constructing optimal frequency-hopping sequence (FHS) sets with low hit zone (LHZ) is presented based on the Cartesian product. By the general structure, we obtain infinitely many optimal LHZ FHS sets with new and flexible parameters by combining the optimal LHZ FHS sets with one-coincidence sequence sets. Moreover, our constructions remove the constraint requiring that the extension factor is co-prime with the length of the original FHSs. In this paper, most of the extension constructions suffer from this constraint. As a result, our constructions allow great flexibility of choosing parameters of the LHZ FHS sets for a given quasi-synchronization frequency-hopping spread spectrum system.

More Optimal Difference Systems of Sets and Frequency-Hopping Sequences From Zero-Difference Functions

Difference systems of sets (DSS) and frequency-hopping sequences (FHS) are two objects with many applications in wireless communication. Zero-difference balanced function (ZDBF) and near zero-difference balanced functions (NZDBF) are two types of functions which can be used to obtain optimal DSSs and FHSs. In order to obtain more optimal DSSs and FHSs, zero-difference function (ZDF) as a generalization of ZDBF and NZDBF was recently proposed. In this paper, four classes of ZDFs with good applications are given from some known ZDBFs. It is noticed that these ZDFs are neither ZDBFs nor NZDBFs. As a result, more optimal DSSs and FHSs with new flexible parameters are obtained.

Improved Singleton bound on frequency hopping sequences and optimal constructions

Frequency hopping (FH) sequences play an important role in FH spread spectrum communication systems. In this paper, a new theoretical bound on the FH sequences with respect to the size of the frequency slot set, the sequence length, the family size, and the maximum periodic Hamming correlation is established. The new bound is tighter than the Singleton bounds on the FH sequences derived by Ding et al. (IEEE Trans Inf Theory 55:3297–3304, 2009) and Yang et al. (IEEE Trans Inf Theory 57:7605–7613, 2011) (first by Sarwate in: Glisic and Leppanen (eds) Code division multiple access communications, Springer, Boston, 1995). In addition, the new bound employes the M $$\ddot{\text {o}}$$ bius function. Then, more optimal FH sequence sets are obtained from Reed–Solomon codes. By utilizing the properties of cyclic codes, a new class of optimal FH sequence sets is obtained whose parameters meet the new bound. Further, two new constructions of FH sequence sets are presented. More new FH sequence sets are obtained by choosing proper base sequence sets. Meanwhile, the FH sequence sets constructed are optimal with respect to the new bound.

Optimal FHSs and DSSs via near zero-difference balanced functions

Zero-difference balanced (ZDB) functions were introduced by Ding in connection with constructions of optimal constant composition codes and optimal and perfect difference systems of sets. Based on such functions, people have constructed optimal constant weight codes and optimal frequency-hopping sequences. In order to obtain more optimal cryptographic objects, the zero-difference balanced (ZDB) function is generalized to the near zero-difference balanced (N-ZDB) function in the present paper, whose characterizations are partially given. Furthermore, we prove that near zero-difference balanced (N-ZDB) functions are equivalent to partitioned almost difference families (PADFs) in design theory. As the main contribution of this paper, three classes of the N-ZDB functions are proposed by means of the partition of Zn, where n is an odd positive integer. Employing these N-ZDB functions, we obtain at the same time optimal frequency-hopping sequences and optimal difference systems of sets with flexible parameters.

Two classes of optimal frequency-hopping sequences with new parameters

Direct-sequence spread spectrum and frequency-hopping (FH) spread spectrum are two main spread-coding technologies. Frequency-hopping sequences (FHSs) achieving the well-known Lempel–Greenberger bound play an important part in FH code-division multiple-access systems. Our objective is to construct more FHSs with new parameters attaining the above bound. In this paper, two classes of FHSs are proposed by means of two partitions of $${{\mathbb {Z}}_{v}}$$ , where v is an odd positive integer. It is shown that all the constructed FHSs are optimal with respect to the Lempel–Greenberger bound. By choosing appropriate injective functions, infinitely many optimal FHSs can be recursively obtained. Above all, these FHSs have new parameters which are not covered in the former literature.

Frame difference families and resolvable balanced incomplete block designs

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational $$(pq+1,p+1,1)$$ -RBIBDs over $${\mathbb {F}}_{p}^+ \times {\mathbb {F}}_{q}^+$$ are derived, and the existence of $$(125q+1,6,1)$$ -RBIBDs is discussed. We construct (v, 8, 1)-RBIBDs for $$v\in \{624,$$ $$1576,2976,5720,5776,10200,14176,24480\}$$ , whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.

Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation

In practice, when a frequency-hopping sequence (FHS) set is applied in a frequency-hopping multiple-access (FHMA) system, its periodic partial Hamming correlation (PPHC) rather than its periodic Hamming correlation (PHC) within the whole period is used to evaluate the system performance. Moreover, FHS sets with low hit zone (LHZ) can be well applied in quasi-synchronous (QS) FHMA systems in which some relative time delay among different users within a zone around the origin can be allowed. Therefore, it is very urgent to conduct research on LHZ FHS sets with optimal PPHC property in depth. In this paper, we first derive a new tighter lower bound on the maximum PPHC of an LHZ FHS set. Then we present a new class of optimal one-coincidence FHS sets. Finally we have a construction of LHZ FHS sets which can be optimal with respect to our new lower bound.

New families of strictly optimal frequency hopping sequence sets

Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHS sets with optimal partial Hamming correlation. We present several direct constructions for balanced nested cyclic difference packings (BNCDPs) and balanced nested cyclic relative difference packings (BNCRDPs) by using trace functions and discrete logarithm. We also show three recursive constructions for FHS sets with partial Hamming correlation, which are based on cyclic difference matrices and discrete logarithm. Combing these BNCDPs, BNCRDPs and three recursive constructions, we obtain infinitely many new strictly optimal FHS sets with respect to the Peng-Fan bounds.

A generic method to construct zero-difference balanced functions

Zero-difference balanced (ZDB) function plays an important role in communication field. In this paper, we propose a generic method to construct ZDB functions on generic algebraic rings. Using this method, we construct many new ZDB functions and retrieve some existing ZDB functions in a much simpler way. Moreover, new applications of the constructed ZDB functions, such as constructing optimal constant weight codes and optimal frequency-hopping sequences, are presented.

Zero-Difference Balanced Functions With New Parameters and Their Applications

As an optimal combinatorial object, zero-difference balanced (ZDB) functions introduced by Ding in 2008, are a generalization of the well-known perfect nonlinear functions. ZDB functions have received much attention in recent years due to its important applications in coding theory and sequence design. One objective of this paper is to present a construction of ZDB functions based on a kind of generalized cyclotomy. It generates ZDB functions over cyclic group with new parameters which can not be produced by earlier constructions. Another objective of this paper is to employ these ZDB functions to obtain at the same time: 1) optimal constant-composition codes; 2) perfect difference systems of sets; and 3) optimal frequency-hopping sequences, all with new parameters.