On the Hamming Auto- and Cross-correlation Functions of a Class of Frequency Hopping Sequences of Length pn p^{n}

In this paper, a new class of frequency hopping sequences (FHSs) of length $p^{n}$ is constructed by using Ding-Helleseth generalized cyclotomic classes of order 2, of which the Hamming auto- and cross-correlation functions are investigated (for the Hamming cross-correlation, only the case $p\equiv 3\pmod 4$ is considered). It is shown that the set of the constructed FHSs is optimal with respect to the average Hamming correlation functions

Construction of three classes of Strictly Optimal Frequency-Hopping Sequence Sets

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 class is based on a generic construction, the second and third classes are based from the trace map

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 (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 $\mathcal{C}$ be a cyclic code of length $n$ over a finite field $\mathbb{F}_q$ such that $\mathcal{C}$ contains the one-dimensional subcode $\mathcal{C}_0=\{(\alpha,\alpha,\cdots,\alpha)\in \mathbb{F}_q^n\,|\,\alpha\in \mathbb{F}_q\}.$ Two codewords of $\mathcal{C}$ 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 $\mathcal{C}\setminus\mathcal{C}_0$ has size $n$. This result addresses an open question raised by Ding {\it et al.} in \cite{Ding09}. 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.Comment: to appear in Designs, Codes and Cryptograph

Asymptotic Gilbert-Varshamov bound on Frequency Hopping Sequences

Given a $q$-ary frequency hopping sequence set of length $n$ and size $M$ with Hamming correlation $H$, one can obtain a $q$-ary (nonlinear) cyclic code of length $n$ and size $nM$ with Hamming distance $n-H$. Thus, every upper bound on the size of a code from coding theory gives an upper bound on the size of a frequency hopping sequence set. Indeed, all upper bounds from coding theory have been converted to upper bounds on frequency hopping sequence sets (\cite{Ding09}). On the other hand, a lower bound from coding theory does not automatically produce a lower bound for frequency hopping sequence sets. In particular, the most important lower bound--the Gilbert-Varshamov bound in coding theory has not been transformed to frequency hopping sequence sets. The purpose of this paper is to convert the Gilbert-Varshamov bound in coding theory to frequency hopping sequence sets by establishing a connection between a special family of cyclic codes (which are called hopping cyclic codes in this paper) and frequency hopping sequence sets. We provide two proofs of the Gilbert-Varshamov bound. One is based on probabilistic method that requires advanced tool--martingale. This proof covers the whole rate region. The other proof is purely elementary but only covers part of the rate region

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 some new algebraic and combinatorial constructions for FHS sets, including an algebraic construction via the linear mapping, two direct constructions by using cyclotomic classes and recursive constructions based on cyclic difference matrices. 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.Comment: 10 page

Construction of Frequency Hopping Sequence Set Based upon Generalized Cyclotomy

Frequency hopping (FH) sequences play a key role in frequency hopping spread spectrum communication systems. It is important to find FH sequences which have simultaneously good Hamming correlation, large family size and large period. In this paper, a new set of FH sequences with large period is proposed, and the Hamming correlation distribution of the new set is investigated. The construction of new FH sequences is based upon Whiteman's generalized cyclotomy. It is shown that the proposed FH sequence set is optimal with respect to the average Hamming correlation bound.Comment: 16 page

A Combinatorial Model of Interference in Frequency Hopping Schemes

In a frequency hopping (FH) scheme users communicate simultaneously using FH sequences defined on the same set of frequency channels. An FH sequence specifies the frequency channel to be used as communication progresses. Much of the research on the performance of FH schemes is based on either pairwise mutual interference or adversarial interference but not both. In this paper, we evaluate the performance of an FH scheme with respect to both group-wise mutual interference and adversarial interference (jamming), bearing in mind that more than two users may be transmitting simultaneously in the presence of a jammer. We establish a correspondence between a cover-free code and an FH scheme. This gives a lower bound on the transmission capacity. Furthermore, we specify a jammer model and consider what additional properties a cover-free code should have to resist the jammer. We demonstrate that a purely combinatorial approach is inadequate against such a jammer, but that with the use of pseudorandomness, we can have a system that has high throughput as well as security against jamming.Comment: 18 pages, submitted to journa

Frequency hopping sequences with optimal partial Hamming correlation

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 FHSs and FHS sets with optimal partial Hamming correlation. We first establish a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property. By virtue of this correspondence, some FHSs and FHS sets with optimal partial Hamming correlation are constructed from various combinatorial structures such as cyclic difference packings, and cyclic relative difference families. We also describe a direct construction and two recursive constructions for FHS sets with optimal partial Hamming correlation. As a consequence, our constructions yield new FHSs and FHS sets with optimal partial Hamming correlation.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1506.0737

Sets of Zero-Difference Balanced Functions and Their Applications

Zero-difference balanced (ZDB) functions can be employed in many applications, e.g., optimal constant composition codes, optimal and perfect difference systems of sets, optimal frequency hopping sequences, etc. In this paper, two results are summarized to characterize ZDB functions, among which a lower bound is used to achieve optimality in applications and determine the size of preimage sets of ZDB functions. As the main contribution, a generic construction of ZDB functions is presented, and many new classes of ZDB functions can be generated. This construction is then extended to construct a set of ZDB functions, in which any two ZDB functions are related uniformly. Furthermore, some applications of such sets of ZDB functions are also introduced.Comment: 20 page

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 $\left(q^m-1,\frac{q^{m-t}-1}{r},rq^t;\frac{q^{m-t}-1}{r}+1\right)$ frequency hopping sequence sets archiving the Peng-Fan bound. Thus, the FHS sets constructed in literatures using either multiplicative groups or additive groups of finite fields are all included in our family. In addition, some other FHS sets can be obtained via the well-known recursive constructions through one-coincidence sequence set