There are two types of bounds on low-hit-zone (LHZ) frequency hopping (FH) sequence sets. One is the lower bound on the maximum Hamming correlation of LHZ FH sequence sets derived by Peng et al. in 2006 (Peng-Fan-Lee bound), and the other is the upper bound on the family size of LHZ FH sequence sets derived by Liu et al. in 2019. Those bounds are not tight in some cases although they can be met in most cases. In this paper, by introducing the Möbius function we derive a new bound on LHZ FH sequence sets which is tighter than that derived by Liu et al.. The new bound deduces the bound on conventional FH sequence sets forZ=N−1, which was derived by Liu et al. in 2014. Further, by establishing an inequality relationship between LHZ FH sequence sets and Krawtchouk polynomials, we derive a linear programming bound on LHZ FH sequence sets, which is sometimes tighter than the Peng-Fan-Lee bound and the upper bound derived by Liu et al.. It is actually a nonlinear programming bound for gcd(Hm+ 1;N) ≠ 1 andZ≥N/gcd(Hm+1;N). By settingZ=N−1, a linear programming bound on conventional FH sequence sets can be deduced which is tighter than that derived by Liu in 2025 for some cases.

Frequency hopping multiple access (FHMA) techniques applied to critical wireless applications often encounter the effects of multiple-access interference (MAI) and follower jamming. The generalized orthogonal FH, which includes no-hit-zone (NHZ) and low-hit-zone (LHZ) FH sequence (FHS), has emerged as a highly promising hopping pattern in quasi-synchronous FHMA systems due to its ability to minimize MAI, but is unable to resist follower jamming attacks. In this paper, we firstly introduce the wide-gap property into an improved NHZ-FHS (i.e., WG-NHZ-FHS), where the frequencies at adjacent time-slots are hopped across a relatively wide gap of bandwidth, so as to address the issues of the MAI and the follower jamming. A general method to generate optimal WG-NHZ-FHS sets satisfying the theoretical bound is proposed. Subsequently, to present the merits of such a hopping pattern, the numerical error-rate performance of the WG-NHZ-FHS set imposed in a quasi-synchronous FHMA system under a follower jamming attacker is derived by using the characteristic function method; in addition, the obtained numerical result is extended to the case of WG-LHZ-FHS as the comparison benchmark. Finally, the numerical analysis and the system simulation results reveal that, compared to the previous types of FHSs, the proposed WG-NHZ-FHS set provides the lower error-rate and higher reliability for critical FHMA applications thanks to its inherent advantages of the MAI-free and the follower jamming suppression.

In order to correct different numbers of erasures in distributed storage systems, the design of locally repairable codes with hierarchical locality (H-LRCs) is crucial. In this paper, we studyh-level H-LRCs wherehis not limited to 2. We present six classes of cyclich-level H-LRCs which are optimal with respect to the generalized Singleton-like bound. The first five classes of cyclich-level H-LRCs have length$ln_{1}$such that$\gcd (l,q)=1$, and$n_{1}|(q-1)$or$n_{1}|(q+1)$. The last class of cyclich-level H-LRCs has length$n|(q+1)$. The minimum Hamming distances of them are$d=\delta _{1}+i$whereican take 0, 2,$\delta _{h}$, and so on. Furthermore, these six classes of cyclich-level H-LRCs have new parameters which are not covered in the literature.

There are several theoretical bounds on frequency hopping (FH) sequences. Each bound is tight in most cases while not tight in some other cases. Besides, the linear programming bound on FH sequences directly converted from that on error correcting codes can be made to be tighter due to the special structure of FH sequences. In this paper, we first give some properties of FH sequences and derive an inequality relationship between FH sequences and Krawtchouk polynomials. By utilizing those properties of FH sequences and the inequality relationship, we establish a linear programming bound on FH sequences. It is actually a nonlinear programming bound for gcd (Hm+1,N)1 and ΣHmj=0Aj≥qHm+1–qHm+1/gcd(Hm+1,N)–1, but not difficult to be solved. It is showed that the linear programming bound is tighter than the Peng-Fan bound (Plotkin bound), the sphere-packing bound, the Singleton bound, and the improved Singleton bound in some cases.

In order to correct different numbers of erasures in distributed storage systems, the design of locally repairable codes with hierarchical locality (H-LRCs) is crucial. In this paper, we construct three classes of optimal cyclic H-LRCs. The minimum Hamming distance of the first class of optimal cyclic H-LRCs is$d=\delta _{1}=\delta +\mu n'+\nu +1$while that of the second class of optimal cyclic H-LRCs is$d=\delta _{1}+1=\delta +\mu n'+\nu +2$for some flexible integers$\delta,\mu,\nu $. The first two classes of optimal cyclic H-LRCs also have unbounded length. Although the length of the third class of optimal cyclic H-LRCs is not unbounded, it can reach large values. These classes of optimal cyclic H-LRCs by our constructions have new and flexible parameters compared with those in the literature.

Sparse code multiple-access (SCMA) is an emerging technique to support massive connectivity in 5G networks and beyond. In SCMA transmissions, some resource-blocks may undergo certain contamination due to deep fading and/or jamming attacks, thus leading to severe performance degradation over such contaminated ones. Besides, the current SCMA infrastructure is normally deployed in single-cell. To deploy the SCMA into multicell networks under contaminated/jamming channels, we propose a novel frequency-hopping based SCMA (FH-SCMA) for quasi-synchronous multi-cell networks, in which the entire subcarrier-channels of every codeword keep hopping over the multiple resource-blocks according certain hopping pattern. We propose and design a pseudo-randomly orthogonal hopping pattern to adapt to the specific requirements of quasi-synchronous FH-SCMA multi-cell networks. Our analysis and simulation results indicate that the proposed FH-SCMA leads to both improved user capacity and error-rate performance, whilst remaining resilient to the inter-cell interference.

In the future, plenty of wireless applications (e.g., Bluetooth, WiFi, Zigbee, and so forth) will be congested in the limited physical resource, which results in the serious interference. In this paper, we exploit a wireless transmission infrastructure to address this issue by occupying the minimum physical resource. Firstly, we introduce an optimal quasiorthogonal frequency-hopping sequence (QO-FHS) set, which has a lower hit-rate but with the less spectral resource than the conventional hopping pattern. Then, a QO-FH based adaptive beam-forming (BF) receiver (QO-FH/BF) is proposed to combat the interference. The theoretical performance of the proposed system in terms of bit-error-rate (BER) is derived as a function of the parameters of QO-FHS set and adaptive BF in the asynchronous access scenario. The numerical and simulation results show that, compared with the conventional FH systems with random hopping pattern, the proposed QO-FH/BF system can remarkably suppress the interference and support the high capacity of network. The anti-interference capability of the proposed QO-FH/BF system depends on the degree of freedom of the combination of QO-FH and BF techniques.

In frequency-hopping sequence (FHS) design, the average Hamming correlation is an important performance indicator. In this paper, the average Hamming correlation of low-hit-zone (LHZ) FHS sets, which has not yet been reported previously, is studied. A lower bound on the average Hamming auto-correlation and the average Hamming cross-correlation of LHZ FHS sets is firstly derived. The new lower bound includes the bound for conventional FHS sets derived by Peng et al. as special case. Then a sufficient and necessary condition for an LHZ FHS set with optimal average Hamming correlation is given. Further, a construction of LHZ FHS sets with optimal average Hamming correlation is presented, whose parameters meet the lower bound on the average Hamming correlation of LHZ FHS sets.

Multi-tier interference is a critical issue in the HetNets uplinks, which dramatically deteriorates the system transmission performance and capacity. In this paper, we propose a prioritized radio-access scheme based on a frequency hopping (FH) technique to address the multi-tier interference issue. To endow the radio-access priorities to the users located at different tiers, a novel FH pattern (FH sequence set) with multi-level Hamming correlations are designed. As a study case, a low peak-to-average-power-ratio (PAPR) FH-based OFDM system (i.e, OFDMlFH system) employing the proposed FH sequence set is investigated in multi-tier HetNet uplinks. The performance in terms of anti-interference capability and spectral efficiency benefited from the proposed FH pattern are presented. The numerical and simulation results show that the proposed FH sequence set can provide a superior capability to mitigate the multi-tier interference in HetNet uplinks; meanwhile guarantee the transmission quality and the spectral efficiency for multi-tier users, even at the cell edge of HetNets.

Anti-interference is a major challenge in wireless communications. In this paper, a new set of optimal no-hitzone (NHZ) frequency-hopping (FH) sequence (so-called NHZ FH pattern) are constructed, which offers lower frequency-hits than other hopping patterns. With such a NHZ FH pattern, an improved FH system using an adaptive beam-forming (BF) receiver (NHZ/FH-BF system) is investigated for the interference mitigation. By combining NHZ/FH and BF techniques, the proposed system can fully harness the advantage of antiinterference from the spatial and the frequency domains to improve the anti-interference capability. The numerical and simulation results show that the anti-interference performance and the spectral efficiency of the proposed NHZ/FH-BF system significantly outperform the traditional systems, i.e., the NHZ/FH system with single antenna and the FH-BF system with the uniformly-distributed FH pattern.

This paper is focused on the optimal design and performance analysis of a novel class of no-hit-zone frequencyhopping sequences (NHZ-FHSs) in quasi-synchronous FH multiple-access (QS-FHMA) systems. Although traditional NHZ-FHSs can offer interference-free FHMA performance when the signal arrival delay (τ) does not exceed the width of NHZ (Znh), i.e., |τ| ≤ Znh, the same performance cannot be guaranteed for |τ| > Znhin which traditional NHZ-FHZs may have large Hamming correlation (which denotes the total number of frequency hits of sequences) and consequently poor error probability performance. Since the strict quasi-synchronization (i.e., |τ| ≤ Znh) may be hard to maintain at all times in practical FHMA networks (e.g., infrastructureless ad hoc networks), it is also important to minimize the Hamming correlation for time-shifts outside of the NHZ. Such FHSs are called strong NHZ-FHSs (SNHZ-FHSs) in this paper. We derive a lower bound on the maximum Hamming correlation outside of the NHZ and then present a design algorithm for the optimal SNHZ-FHS set meeting our proposed lower bound with equality. We analyze the bit-error-rate (BER) performance of the FHMA system employing the proposed SNHZ-FHS sets using the average Hamming correlation function. The theoretical analysis and simulation results show that the proposed optimal SNHZ-FHS sets are feasible for practical FHMA networks with relaxed timing requirement and enhanced BER performance.

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 > q2/M and q ≥ 2, and that by Cai et al. when N > q2/M and q ≥ 2N/(N - 2). Furthermore, we define a special partition-type difference packing (DP) called [N, ∇, Hal] PDP and give several classes of [N, ∇, Hal] 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, ∇, Hal] PDPs, which lead to infinitely many classes of strictly optimal FH sequences with new parameters. Moreover, based upon an [N, ∇, Hal] 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.

The performance of frequency-hopping multiple-access (FHMA) systems is mainly determined by the FH sequence set (FHS). Constructing the FHS with optimal Hamming cross- and auto-correlations under the varying access-delay is a major challenge for FHMA systems to achieve the superior performance. In this paper, we propose a novel kind of no-hit zone FH sequence (NHZ-FHS), which has the orthogonality for small access-delay and the near-orthogonality for large access-delay. A design algorithm of the new NHZ-FHS is presented based on the traditional FH sequences. Then, as a case study of FHMA system with M-ary frequency-shift-keying (MFSK/FHMA), the multiple-access interference (MAI) rejection benefited from the proposed FHS are analyzed, and the decision variable is derived as a function of the Hamming correlation, which is utilized for Monte Carlo simulation. The simulation and numerical results show that the proposed NHZ-FHS can completely avoid MAI for quasi-synchronous FHMA system; meanwhile it can provide the lower MAI and better performance for synchronous system, compared with the exiting pseudo-random FHSs. The proposed NHZ-FHS relaxes the requirement of strict synchronization mechanism, which is more suitable for the FHMA network deploying under the poor communication conditions.