Within the zero-correlation-zone (ZCZ), ZCZ sequence sets exhibit ideal correlation properties, which is highly advantageous for both wireless communications and radar sensing applications. Recently, to achieve optimal training for spatial modulation (SM), Paiet al. introduced the concept of sparse ZCZ (SZCZ) sequence sets, where each column contains only one non-zero element. In this paper, we first extend the SZCZ sequence set concept by permitting multiple non-zero elements per column, thereby accommodating training design requirements for generalized SM (GSM) systems. Then, we propose a direct construction of SZCZ sequence sets with parameter (qn+k,qm+n+k, (q−1)qπ(2)−1+(q−2)qπ(3)−1, (qn,qm+n)) based on restricted extended Boolean functions. Compared to existing works, the proposed SZCZ sequence sets exhibit a greater ZCZ width and can be applied to both SM and GSM training designs simultaneously. Simulation results show that compared with other training sequences, the channel estimation performance is significantly improved when the proposed SZCZ sequence set is used as the training sequences.

Arrays with good correlation properties have emerged with promising applications in wireless communication, including ultra-wideband (UWB), two-dimensional (2- D) synchronization, and massive multiple-input multiple-output (MIMO). In this paper, we propose two efficient algorithms for designing arrays with low autocorrelation by minimizing the weighted integrated sidelobe level (WISL) and complementary ISL (CISL), respectively. We formulate these metrics as non-convex quartic problems under unimodular constraints in the frequency domain. Using the majorization-minimization (MM) framework, these problems are simplified into quadratic forms, ensuring monotonic convergence to a stationary point. Furthermore, a projection algorithm is introduced to relax unimodular constraints to peak-to-average power ratio (PAPR) constraints, enhancing the correlation performance of the optimized arrays. All proposed algorithms can be efficiently implemented using two-dimensional FFT/IFFT operations. Furthermore, a quasi-complementary array set (QCAS), generated via the proposed MM-ACISL algorithm, is applied as an omnidirectional precoding matrix in massive MIMO systems. Numerical experiments demonstrate that the proposed algorithms outperform existing methods in terms of both correlation performance and computational efficiency. Moreover, the resulting QCAS scheme effectively enables omnidirectional transmission in MIMO systems.

Low ambiguity zone (LAZ) sequences play a crucial role in modern integrated sensing and communication (ISAC) systems. In this paper, we introduce a novel class of functions known as locally perfect nonlinear functions (LPNFs). By utilizing LPNFs and interleaving techniques, we propose three new classes of both periodic and aperiodic LAZ sequence sets with flexible parameters. The proposed periodic and aperiodic LAZ sequence sets are asymptotically optimal with respect to the periodic and aperiodic lower AF bounds, respectively, which were proposed recently in [IEEE J. Sel. Areas Commun. 40 (6): 1809-1822]. Notably, the aperiodic LAZ sequence sets are the first such sequence sets in the literature that satisfy the bound. Finally, we demonstrate that the proposed sequence sets are cyclically distinct.

Sequences with low ambiguity zone (LAZ) properties are employed in integrated sensing and communication (ISAC) systems. In this letter, we introduce a new class of LAZ sequence sets based on cubic sequences. The proposed LAZ sequence set is asymptotically optimal with respect to the Ye-Zhou-Fan-Liu-Lei-Tang (YZFLLT) bound when the sequence length is odd. Otherwise, it achieves$\sqrt{2}$times of the YZFLLT bound. Furthermore, the sequence sets provide flexible choices for the sequence lengths, set size, and the LAZ region.

This paper studies Doppler resilient complementary sequences (DRCSs) whereby the ambiguity functions (AFs) of multiple element sequences are summed to attain low/zero AF values. We first derive a set of AF lower bounds for unimodular DRCS sets, which include the existing bounds on AFs as special cases. These bounds may be used as theoretical design guidelines to measure the optimality of DRCS sets against Doppler effect. In addition, we introduce some constructions of DRCS sets based on mathematical tools such as orthogonal matrices, circular Florentine rectangles and difference sets, which can generate the optimal DRCS set. Finally, we evaluate the feasibility of DRCSs for pulse train waveform design.

Frequency hopping sequences (FHSs) play a crucial role in frequency hopping (FH) communication systems due to their strong anti-interference ability, low interception probability, high confidentiality and strong concealment. The objective of this paper is to construct FHSs for quasi-synchronous frequency-hopping multiple access (FHMA) communication systems that simultaneously achieve optimal no-hit zone (NHZ) length and optimal gap. To accomplish this, the paper first derives tighter upper bounds for the gap size in both periodic and aperiodic scenarios under the assumption that all frequencies within the designated frequency slot set are fully utilized. Subsequently, this paper proposes a class of wide-gap frequency hopping sequences (WGFHSs) and a class of multi-timeslot wide-gap frequency hopping sequences (MTWGFHSs), both of which simultaneously exhibit optimal NHZ length and optimal gap.

In this paper, a general class of polyphase sequences called Generalized Zadoff-Chu (GZC) sequences is presented, which is based on a quadratic function with real-valued coefficients. The conventional ZC sequences, P3, and P4 codes are included as special cases of GZC sequences. It is shown that, with the help of grid search algorithm, the optimized GZC sequences have significantly lower PMEPR than the well-known sequences such as Golay sequences, m-sequences, P3 and P4 codes, and ZC sequences. Numerical simulations suggest that the minimum PMEPR tends to a small constant of less than 1.43 dB as the sequence length approaches infinity.

Complete complementary codes (CCCs) have important applications in communication, radar, and information security. In modern communication and radar systems, certain spectrum is reserved or prohibited from transmission, which leads to the so-called spectrally null constrained (SNC) problem. Compared with vast works on conventional CCCs, relatively little is known about SNC-CCCs. One objective of this paper is to derive several constructions of CCCs with more flexible settings from the graph of extended Boolean functions. This generalizes earlier achievements on CCCs from recent literature. Another objective of this paper is to employ the graphs of extended Boolean functions and polynomial representation of sequences to construct SNC-CCCs.

Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase and end-shift positions and zero cross-correlation sums around the end-shift positions. CZCPs can be crucial in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency-selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio (CZCR). A construction framework of CZCPs with a large ZCZ ratio is proposed using Turyn’s method on some seed CZCPs and GCPs. By choosing suitably the seed CZCPs, we obtain 24 classes of new CZCPs with large CZCR. Especially, if the GCP is strengthened, our resultant CZCPs have the maximum$\mathrm {\mathbf{CZCR}}\approx \frac {M-1}{M}$for$M\in \{6,12,24,28,48,56\}$. We also obtain optimal CZCPs with new parameters (28, 13), (48, 23), (56, 27), (96, 47) and (112, 55), which can be extended to$(96N,47N)$-CZCPs and$(112N,55N)$-CZCPs respectively for any Golay number$N$.

Massive machine type communication (mMTC) is one of the core components of 6G communication systems to fulfil the demand of massive connectivity of billions of Internet-of-Things (IoT) devices. Due to its various advantages, grant-free random access schemes are considered as a promising technique to implement mMTC. The key to grant-free random access is active device detection at the base station. In this paper, firstly we introduce the odd total squared correlation (OTSC) of binary signature sets, then derive a corresponding lower bound. Systematic constructions of optimal signature sets based on the known odd periodic complementary sets (OPCS), almost binary sequences, large Kasami subsets and ideal sequences are presented, which are all optimal with respect to the derived OTSC lower bound. Next, it is demonstrated that the optimal OTSC signature sets can be effectively used in massive device activity detection. Using efficient approximate message passing with minimum mean squared error (AMP-MMSE) algorithm, it is shown that the sensing matrices arising from OTSC-optimal signature sets and periodic total squared correlation (PTSC)-optimal signature sets performs better than the popular Gaussian random matrix.

Z-complementary code sets (ZCCSs) can support interference-free multi-carrier code-division multiple access (MC-CDMA) in quasi-synchronous channels due to their favourable correlation properties. In this letter, based on the existing optimal ZCCSs and CCCs, we propose new class of optimal ZCCSs using Kronecker product. The resultant sequence sets have enlarged set size and sequence lengths which have not been reported before.

In frequency hopping (FH) communication systems, frequency hopping sequences (FHSs) are crucial in determining the system’s anti-jamming performance. If FHSs can ensure a wide-gap between two adjacent frequency points to avoid the frequency points with high interference probability, it will significantly improve the FH communication system’s anti-interference ability. Moreover, if each frequency point appears at the same number of times in a sequence period, the system’s anti-electromagnetic interference will be enhanced. Therefore, it is desirable to employ FHSs with low Hamming autocorrelation, wide frequency-hopping gap, and good uniformity in practical applications. However, to the best of our knowledge, no such infinite classes of FHSs have been reported in the literature to date. This paper aims to present two constructions of uniform wide-gap frequency-hopping sequences (WGFHSs) by concatenating two or three adequately designed sequences. For the first time, we obtain two infinite classes of WGFHSs, which are optimal with respect to the well-known Lempel-Greenberger bound.

Due to their beautiful aperiodic correlation properties and low peak-to-average power ratio (PAPR), Golay complementary pairs (GCPs) and complementary sequence sets (CSSs) have been well studied. However, conventional GCPs and CSSs can only adapt to contiguous spectral resource allocation in wireless communication systems, and it is difficult for them to be compatible with non-contiguous resource allocation. Inspired by recent constructions of non-contiguous GCPs given by Şahin and Yang, we propose a new construction of non-contiguous GCPs, which can be regarded as an extension of Turyn’s construction. Besides, we then propose some constructions of non-contiguous CSSs, which can adapt to flexible resource allocation, having more flexible lengths, and lower time domain cross correlation than non-contiguous GCPs. Finally, we propose a CSS-based physical uplink control channels (PUCCH) scheme for up to 2 uplink control information (UCI) bits. Simulations show that its performance is similar to the GCP-based PUCCH scheme.

A pair of sequences of lengthLis called a Z-complementary pair (ZCP) with zero correlation zone (ZCZ)Zif the sequences have zero aperiodic autocorrelation sums at eachof the non-zero time-shifts smaller thanZ. Such a pair is denoted byan(L,Z)-ZCP for short. In this letter, we propose a construction ofZCPs via properly cascading sequences from a Golay complemen-tary pair (GCP). The proposed construction leads to(3N,2N)-ZCPs,(5N,3N)-ZCPs,(7N,4N)-ZCPs,(9N,5N)-ZCPs,(11N,6N)-ZCPs,(12N,10N)-ZCPs,(13N,7N)-ZCPsand(14N,12N)- ZCPs, whereNis the length of the chosen GCP. Itturns out that the resultant ZCPs have low peak-to-mean envelopepower ratios.

Recently, mutually orthogonal complementary sequence sets (MOCSSs) have been found many important applications in communication systems, radar, etc. Most of the known constructions of MOCSSs are based on generalized Boolean functions (GBFs) and hence mostly have lengths of power-of-two. A few constructions of MOCSSs are also based on paraunitary (PU) matrices and hence mostly have power-of-an-integer lengths. The objective of this paper is to develop a general framework to construct more MOCSSs consisting of sequences with non-power-of-two lengths. The proposed framework is based on complete complementary codes and even-shift complementary sequence sets.

One objective of this paper is to propose a novel class of sequence pairs, called “quasi-orthogonal Z-complementary pairs (QOZCPs)”, each depicting Z-complementary property for their aperiodic auto-correlation sums and also having a low correlation zone when their aperiodic cross-correlation is considered. Construction of QOZCPs based on Successively Distributed Algorithms under Majorization Minimization (SDAMM) is presented. Another objective of this paper is to apply the proposed QOZCPs in fully polarimetric radar systems and analyse the corresponding ambiguity functions. It turns out that QOZCP waveforms are much more Doppler resilient than the known Golay complementary waveforms.

The construction of complementary sets (CSs) of sequences with different set size and sequence length become important due to its practical application for OFDM systems. Most of the constructions of CSs, based on generalized Boolean functions (GBFs), are of length 2α(α is a natural number). Recently some works have been reported on construction of CSs having lengths non-power of two, i.e., in the form of 2m-1+ 2v(m is natural number, 0 ≤ v <; m), N + 1 and N + 2, where N is a length for which q-ary complementary pairs exist. In this letter, we propose a construction of CSs of lengths M + N for set size 4n, using concatenation of CSs of lengths M and N, and set size 4n, where M and N are lengths for which q-ary complementary pairs exists. Also, we construct CSs of length M + P for set size 8n by concatenating CSs of lengths M and P, and set size 8n, where M and P are lengths for which q-ary complementary pairs and complementary sets of size 4 exists, respectively. The proposed constructions cover all the previous constructions as special cases in terms of lengths and lead to more CSs of new sequence lengths which have not been reported before.

Complementary set sequences (CSSs) are useful for dealing with the high peak-to-average power ratio (PAPR) problem in orthogonal frequency division multiplexing (OFDM) systems. In practical OFDM transmission, however, certain sub-carriers maybe reserved and/or prohibited to transmit signals, leading to the so-called spectral null constraint (SNC) design problem. For example, the DC sub-carrier is reserved to avoid the offsets in D/A and A/D converter in the LTE systems. While most of the current research focus on the design of low PAPR CSSs to improve the code-rate, few works address the aforementioned SNC in their designs. This motivates us to investigate CSSs with SNC as well as low PAPR property. In this article, we present systematic constructions of CSSs under SNCs and low PAPR. First, we show that mutually orthogonal complementary sets (MOCSs) can be used as seed sequences to generate new CSSs with SNC and low PAPR, and then provide an iterative technique for the construction of MOCSs which can be further used to generate complementary sets (CSs) with low PAPRs and spectral nulls at varying positions in the designed sequences. Next, inspired by a recent idea of Chen, we propose a novel construction of these seed MOCSs with non-power-of-two lengths from generalized Boolean functions.

An ambiguity function is widely used in radar and communication. It is highly desirable that an ambiguity function should have low range- and Doppler- sidelobe as the good resolution in both range and Doppler domain. In this paper, a method which combines time-domain binomial design and frequency-domain binomial design via a Point-wise Minimum Processor (PMP) is proposed. The time-domain binomial design is designed by [7] to clear the range sidelobe in a very large Doppler area and the frequency-domain binomial design is proposed by this paper to avert the Doppler sidelobe in a very large range area. It is shown that the new method produces a "thumbtack-like" ambiguity function with a good range/doppler sidelobe suppression and results in good distinction of two moving targets.

A pair of sequences is called odd-length binary Z-complementary pair (OB-ZCP) if it is of odd-length and has zero aperiodic autocorrelation sums (AACSs) for all time-shifts within a certain region around the in-phase position, commonly known as zero correlation zone (ZCZ). There are two types of OB-ZCPs, namely Type-I OB-ZCPs and Type-II OB-ZCPs. Type-I OB-ZCPs have ZCZ around the in-phase position. Type-II OB-ZCPs have the ZCZ around the end-shift position. An OB-ZCP (Type-I or Type-II) of odd-length N is called Z-optimal if it achieves a maximum ZCZ width of (N + 1)/2. To date, a systematic construction of Type-II Z-optimal OB-ZCPs exist only for very limited lengths of the form 2m± 1, where m is a positive integer. It employs insertion method and delete method on binary Golay complementary pairs (GCPs) of length 2mderived from second order Reed-Muller codes. In this article, based on iterative insertion method, we construct Type-II Z-optimal OB-ZCPs of lengths 2m+ 3.

Binary sequences with low odd correlation have important applications in communication systems to reduce interference. In this paper, using the interleaving technique, we present a generic connection between binary sequences with low odd correlation and quaternary sequences with low even correlation. As a result, some new binary sequences with optimal odd auto-correlation magnitude are obtained. Besides, two sets consisting of 2n+1 binary sequences of period 2(2n-1) with the maximum odd correlation magnitude 2((n+1)/2)+ 2 are derived, which are the first two optimal classes of binary sequence sets achieving the Sarwate bound on the odd correlation magnitude in the literature.

Constant composition codes (CCCs) are a special class of constant-weight codes. They include permutation codes as a subclass. The construction of CCCs with parameters achieving certain bounds has been an interesting research topic in coding theory. Recently, Ding established a bridge from zero difference balanced (ZDB) functions to CCCs with parameters meeting the Luo-Fu-Vinck-Chen bound. This provides a new approach for obtaining optimal CCCs. One objective of this paper is to present two new classes of ZDB functions whose parameters have been not covered in the literature. Another objective of this paper is to introduce two classes of CCCs meeting the Luo-Fu-Vinck-Chen bound from these new ZDB functions.

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.

Sequences with good odd periodic autocorrelation property are of importance in applications. In this paper, we presented a lower bound of the magnitude of the odd periodic autocorrelation of binary sequences, and by using Parker's transformation, constructed new binary sequences with odd periodic autocorrelation magnitude achieving the lower bound.

In this paper, we give a lower bound on odd periodic total squared correlation (OPTSC for short) of binary signature sets, which indicates that odd periodic complementary sets and PTSC-optimal signature sets of odd period can be used to design optimal OPTSC signature sets which achieve the new lower bound. Besides, we give three kinds of PTSC-optimal signature sets from ideal sequences and large Kasami subsets.

Frequency-hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, lower bounds on the partial Hamming correlation of FHSs and FHS sets are proposed. They slightly improve the known bounds by Eun et al. and Zhou et al. A construction of FHSs and FHS sets having optimal partial Hamming correlation with respect to the improved bounds is also presented based on the theory of generalized cyclotomy. Our construction yields optimal FHSs and FHS sets with new and flexible parameters not covered in this paper.

Sequences with desirable correlation properties have wide applications in today's communication systems. In this paper, we first extend the known results on zero autocorrelation zone of Golay sequences to 4q-QAM Golay sequences and show three constructions of 4q-QAM Golay sequences with a large zero periodic autocorrelation zone, where q ≥ 2 is an arbitrary integer. We then determine the Golay pairs which have large zero periodic crosscorrelation zone.

Golay sequences have been studied for more than five decades since Golay first discovered those sequences. However, the periodic autocorrelation of a single Golay sequence is unknown. In this paper, for H≥ 2 being an arbitrary even integer, we show there exist three different constructions of H-ary Golay sequences with a zero autocorrelation zone (ZACZ) of length approximately an half, a quarter or one eighth of their period. Those new discoveries on Golay sequences can be explored during synchronization and detection at the receiver end and thus improve the performance of the communication system. We present the application of binary Golay sequences with ZACZ for intersymbol interference (ISI) channel estimation. Compared with m-sequences, Golay-sequence-aided channel estimation has perfect autocorrelations within the zone. Compared with Frank-Zadoff-Chu sequences, Golay-sequence-aided channel estimation requires much lower hardware and computational complexity. We also discuss the performance of Golay-sequence-aided channel estimation in terms of its error variance. Finally, simulations are conducted to show the performance of our proposed scheme against m-sequences and FZC sequences in terms of symbol error rate. The simulations also confirm with our theoretical results.

In this paper, a new construction of zero-difference balanced functions defined on is given, where is an odd positive integer. Based on the generic constructions proposed by Ding, optimal constant composition codes and perfect difference systems of sets with new parameters can be generated from the zero-difference balanced functions constructed in this paper.

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.

Sequences with good correlation properties have been widely used in modern communications, radar and sonar applications. In this paper, we consider the autocorrelation property of the single binary and quaternary Golay sequences, with special focus on their zero periodic autocorrelation zone. Some examples along with one case of the proof will be given in order to demonstrate this zero periodic autocorrelation zone. This finding on Golay sequences can be explored during synchronization and detection at the receiver end and thus improve the performance of the orthogonal frequency-division multiplexing (OFDM) system where Golay sequences are employed for the peak-to-average power reduction.

Sequences with good correlation properties have been widely adopted in modern communications, radar and sonar applications. In this paper, we present that a single H-ary Golay sequence or 4q-QAM Golay sequence has a large zone of zero odd periodic autocorrelation, where H ≡ 0 (mod 4) is a positive integer and q ≥ 2 is an arbitrary integer. The conditions on the permutations employed in the boolean functions are the same as those for the sequences with a large zone of zero (even) periodic autocorrelation. More importantly, sequences with large odd periodic autocorrelations centered around the origin could be used to reduce the multipath interference at the receiver end and thus improve the performance of the communication system.

A Gaussian integer is a complex number whose real and imaginary parts are both integers. A Gaussian integer sequence is called perfect (odd perfect) if the out-of-phase values of the periodic (odd periodic) autocorrelation function are equal to zero. In this letter, for any odd prime p, using the cyclotomic classes of order 2 and 4 with respect to GF(p), we propose perfect and odd perfect Gaussian integer sequences of length p. Several examples are also given.

In this paper, we propose an interleaving construction of new sets of frequency hopping sequences from the known ones. By choosing suitable known optimal frequency hopping sequences and sets of frequency hopping sequences and then recursively applying the proposed construction, optimal frequency hopping sequences and sets of frequency hopping sequences with new parameters can be obtained.

Zero-difference balanced (ZDB) functions were introduced recently by Ding for the construction of optimal constant-composition codes, and optimal and perfect difference systems of sets. They are closely related to partitioned difference families. In this paper, we present generic constructions of ZDB functions from functions with difference-balanced property. In particular, two classes of ZDB functions with new and flexible parameters are reported. Employing these new ZDB functions, we obtain at the same time optimal (1) constant-composition codes, (2) constant-weight codes, and (3) perfect difference systems of sets, all with new and flexible parameters.

In this paper, an upper bound for a hybrid incomplete exponential sum over finite fields is derived. This bound is then used to obtain lower and upper bounds for aperiodic Hamming correlation of frequency-hopping sequences based on power functions.

In this paper, we derive a new bound on maximum nontrivial Hamming correlation of frequency hopping (FH) sequences from the Singleton bound in error correcting code literature, and we discuss the relation between the new bound and the known ones on FH sequences. Further, we construct two classes of FH sequences from punctured Reed–Solomon codes and one class of FH sequences from polynomial functions, which meet the new bound.

In communication systems, frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are used in FH-CDMA systems. In this paper, an earlier idea of constructing optimal sets of frequency hopping sequences is further investigated. New optimal parameters of sets of frequency hopping sequences are obtained with subcodes of the Reed-Solomon codes. Optimal sets of frequency hopping sequences are constructed with a class of irreducible cyclic codes. As a byproduct, the weight distribution of a subclass of irreducible cyclic codes is determined.