Optimal Frequency Hopping Sequence Sets 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>

Construction of Optimal Frequency Hopping Sequence Set with Low-Hit-Zone.

In quasi-synchronous frequency-hopping multiple access (QS-FHMA) systems, low-hit-zone (LHZ) frequency-hopping sequence (FHS) sets have been well-applied to reduce mutual interference (MI). In this paper, we propose three constructions of LHZ FHS sets with new parameters via interleaving techniques. The obtained sequences can be verified that they are optimal with respect to the Peng-Fan-Lee bound.

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.

A Construction of Optimal Frequency Hopping Sequence Set via Combination of Multiplicative and Additive Groups of Finite Fields

In literatures, there are various constructions of frequency hopping sequence (FHS for short) sets with good Hamming correlations. Some papers employed only multiplicative groups of finite fields to construct FHS sets, while other papers implicitly used only additive groups of finite fields for construction of FHS sets. In this paper, we make use of both multiplicative and additive groups of finite fields simultaneously to present a construction of optimal FHS sets. The construction provides a new family of optimal $\left ({{q}^{{m}}-1,\frac {{q}^{{m}-{t}}-1}{{r}},{\it{ rq}}^{{t}};\frac {{q}^{{m}-{t}}-1}{{r}}+1}\right)$ frequency hopping sequence sets archiving the Peng-Fan bound. Thus, some FHS sets constructed in literatures using either multiplicative groups or additive groups of finite fields are included in our family. In addition, some other FHS sets can be obtained via the well-known recursive construction through one-coincidence sequence set.

Simplified Bounds on FHSs Set and Its Strictly Optimal Construction

Frequency-hopping sequences (FHSs) with favorite partial Hamming correlation properties are intensively needed in many synchronization and multiple-access systems. Strictly optimal FHS sets are a kind of FHS sets which has optimal Hamming correlation for any correlation window. In this paper, firstly we present simplified representations of the generalized Lempel–Greenberger bound on the partial Hamming autocorrelation of an FHS and the generalized Peng-Fan bound on the partial Hamming correlation of an FHS set, respectively. Secondly, we propose a direct construction of strictly optimal FHS sets, which interprets the previous construction proposed by Cai, Zhou, Yang and Tang. By choosing appropriate parameters and bijections, we present more flexible constructions of strictly optimal FHS sets.

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.

A new family of optimal FHS sets with composite lengths

In frequency-hopping spread spectrum systems, frequency-hopping sequences (FHSs) are widely used to alleviate the interferences caused by the hits of frequencies. In this paper, based on the Zeng–Cai–Tang–Yang cyclotomy and the Chinese Remainder Theorem, a family of FHS sets with composite lengths is constructed and the Hamming correlations of the new FHS sets are derived by some basic properties of cyclotomic numbers. The results show that the proposed FHS sets are optimal with respect to the Peng–Fan bound. Furthermore, it generalizes some earlier cyclotomic constructions of optimal FHS sets and produces optimal FHS sets with 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 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.

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.

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.

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.

Constructions of optimal low-hit-zone frequency hopping sequence sets

In recent years, the study relating to low-hit-zone frequency hopping sequence sets, including the bounds on the Hamming correlations within the low hit zone and the optimal constructions, has become a new research area attracting the attention of many related researchers. In this paper, two constructions of optimal frequency hopping sequence sets with low hit zone have been employed, one of which is based on m-sequence while the other is based on the decimated sequences of m-sequence. Moreover, in the special case of \(k=n-1\), the construction based on the decimated sequences of m-sequence also yields low-hit-zone frequency hopping sequence sets with optimal periodic partial Hamming correlation property.

New Families of Optimal Frequency Hopping Sequence Sets

Frequency hopping sequences (FHSs) are employed to mitigate the interferences caused by the hits of frequencies in frequency hopping spread spectrum systems. In this paper, we present combinatorial constructions for FHS sets, including direct constructions by using cyclotomic classes, recursive constructions based on cyclic difference matrices, merging blocks, and discrete logarithm. By these constructions, a number of series of new FHS sets are then produced. These FHS sets are optimal with respect to the Peng–Fan bounds.

Recursive construction of optimal frequency‐hopping sequence sets

In this study, first the authors present a simplified representation of the Peng–Fan bounds on the periodic Hamming correlation of frequency-hopping sequence (FHS) sets, which may also be used to check the optimality of an FHS set with respect to the Peng–Fan bounds. Second, they propose a recursive construction of FHS sets from the known ones using some injective functions and the Chinese remainder theorem. It generalises the previous construction of optimal FHSs and FHS sets with composite lengths employing a given function. Without the limit of the specific function, their construction can produce new optimal FHSs and FHS sets that cannot be produced by the earlier construction. By choosing appropriate injective functions and known optimal FHSs and FHS sets, infinitely many new optimal FHSs and FHS sets can be recursively obtained.

Optimal Frequency-Hopping Sequence Sets Based on Cyclotomy

In this paper, a kind of generalized cyclotomy with respect to the square of a prime is presented and the properties of the corresponding generalized cyclotomic numbers are investigated. Based on the generalized cyclotomy, a class of frequency-hopping sequence (FHS) set is constructed. By means of some basic properties of the generalized cyclotomy, we derive the Hamming correlation distribution of the new set. The results show that the proposed FHS set is optimal with regard to the average Hamming correlation (AHC) bound. By choosing suitable parameters, the construction also leads to the optimal FHS set and the optimal FHSs with regard to the maximum Hamming correlation (MHC) bound and Lempel-Greenberger bound, respectively.

Three new classes of optimal frequency-hopping sequence sets

The study of frequency-hopping sequences (FHSs) has been focused on the establishment of theoretical bounds for the parameters of FHSs as well as on the construction of optimal FHSs with respect to the bounds. Peng and Fan (IEEE Trans Inf Theory 50(9):2149---2154, 2004) derived two lower bounds on the maximum nontrivial Hamming correlation of an FHS set, which is an important indicator in measuring the performance of an FHS set employed in practice. In this paper, we obtain two main results. We study the construction of new optimal frequency-hopping sequence sets by using cyclic codes over finite fields. Let $$\mathcal {C}$$C be a cyclic code of length n over a finite field $$\mathbb {F}_q$$Fq such that $$\mathcal {C}$$C contains the one-dimensional subcode $$ \mathcal {C}_0=\{(\alpha ,\alpha ,\ldots ,\alpha )\in \mathbb {F}_q^n\,|\,\alpha \in \mathbb {F}_q\}. $$C0={(ź,ź,ź,ź)źFqn|źźFq}. Two codewords of $$\mathcal {C}$$C are said to be equivalent if one can be obtained from the other through applying the cyclic shift a certain number of times. We present a necessary and sufficient condition under which the equivalence class of any codeword in $$\mathcal {C}\setminus \mathcal {C}_0$$C\C0 has size n. This result addresses an open question raised by Ding et al. (IEEE Trans Inf Theory 55(7):3297---3304, 2009). As a consequence, three new classes of optimal FHS sets with respect to the Singleton bound are obtained, some of which are also optimal with respect to the Peng---Fan bound at the same time. We also show that the two Peng---Fan bounds are, in fact, identical.

Strictly Optimal Frequency-Hopping Sequence Sets With Optimal Family Sizes

Frequency-hopping sequences (FHSs) with favorable partial Hamming correlation properties are desirable in many synchronization and multiple-access systems. An FHS set is said to be strictly optimal if it has optimal partial Hamming correlation for any correlation window. In this paper, we derive upper bounds on the family sizes of FHS sets with respect to partial Hamming correlation from some classical bounds on error-correcting codes. We then present strictly optimal FHS sets having optimal family sizes with respect to one of the new bounds. In particular, our construction gives new parameters not covered in the literature.