In this survey we summarize properties of pseudorandomness and non-randomness of some number-theoretic sequences and present results on their behaviour under the following measures of pseudorandomness: balance, linear complexity, correlation measure of order k, expansion complexity and 2-adic complexity. The number-theoretic sequences are the Legendre sequence and the two-prime generator, the Thue-Morse sequence and its sub-sequence along squares, and the prime omega sequences for integers and polynomials.

Human motion prediction has always been an interesting research topic in computer vision and robotics. It means forecasting human movements in the future conditioning on historical 3-dimensional human skeleton sequences. Existing predicting algorithms usually rely on extensive annotated or non-annotated motion capture data and are non-adaptive. This paper addresses the problem of few-frame human motion prediction, in the spirit of the recent progress on manifold learning. More precisely, our approach is based on the insight that achieving an accurate prediction relies on a sufficiently linear expression in the latent space from a few training data in observation space. To accomplish this, we propose Regressive Gaussian Process Latent Variable Model (RGPLVM) that introduces a novel regressive kernel function for the model training. By doing so, our model produces a linear mapping from the training data space to the latent space, while effectively transforming the prediction of human motion in physical space to the linear regression analysis in the latent space equivalent. The comparison with two learning motion prediction approaches (the state-of-the-art meta learning and the classical LSTM-3LR) demonstrate that our GPLVM significantly improves the prediction performance on various of actions in the small-sample size regime.

In frequency hopping communication, time delay and Doppler shift incur interference. With the escalating upgrading of complicated interference, in this paper, the time-frequency two-dimensional (TFTD) partial Hamming correlation (PHC) properties of wide-gap frequency-hopping sequences (WGFHSs) with frequency shift are discussed. A bound on the maximum TFTD partial Hamming auto-correlation (PHAC) and two bounds on the maximum TFTD PHC of WGFHSs are got. Li-Fan-Yang bounds are the particular cases of new bounds for frequency shift is zero.

In the massive machine-type communication scenario, aiming at the problems of active user detection and channel estimation in the grant-free non-orthogonal multiple access (NOMA) system, new sets of non-orthogonal spreading sequences are proposed by using the zero/low correlation zone sequence set with low correlation among multiple sets. The simulation results show that the resulting sequence set has low coherence, which presents reliable performance for channel estimation and active user detection based on compressed sensing. Compared with the traditional Zadoff-Chu (ZC) sequences, the new non-orthogonal spreading sequences have more flexible lengths, and lower peak-to-average power ratio (PAPR) and smaller alphabet size. Consequently, these sequences will effectively solve the problem of high PAPR of time domain signals and are more suitable for low-cost devices in massive machine-type communication.

In quasi-synchronous FH multiple-access (QS-FHMA) systems, no-hit-zone frequency-hopping sequences (NHZ-FHSs) can offer interference-free FHMA performance. But, outside the no-hit-zone (NHZ), the Hamming correlation of traditional NHZ-FHZs maybe so large that the performance becomes not good. And in high-speed mobile environment, Doppler shift phenomenon will appear. In order to ensure the performance of FHMA, it is necessary to study the NHZ-FHSs in the presence of transmission delay and frequency offset. In this paper, We derive a lower bound on the maximum time-frequency two-dimensional Hamming correlation outside of the NHZ of NHZ-FHSs. The Zeng-Zhou-Liu-Liu bound is a particular situation of the new bound for frequency shift is zero.

Pseudorandom sequences with large linear complexity can resist the linear attack. The trace representation plays an important role in analysis and design of pseudorandom sequences. In this letter, we present the construction of a family of new binary sequences derived from Euler quotients modulo pq, where pq is a product of two primes and p divides q-1. Firstly, the linear complexity of the sequences are investigated. It is proved that the sequences have larger linear complexity and can resist the attack of Berlekamp-Massey algorithm. Then, we give the trace representation of the proposed sequences by determining the corresponding defining pair. Moreover, we generalize the result to the Euler quotients modulo pmqn with m≤n. Results indicate that the generalized sequences still have high linear complexity. We also give the trace representation of the generalized sequences by determining the corresponding defining pair. The result will be helpful for the implementation and the pseudorandom properties analysis of the sequences.

We consider both-ends-fixed k-ary necklaces and enumerate all such necklaces of length n from the viewpoints of symbolic dynamics and β-expansions, where n and k(≥ 2) are natural numbers and β(> 1) is a real number. Recently, Sawada et al. proposed an efficient construction of k-ary de Bruijn sequence of length kn, which for each n ≥ 1, requires O(n) space but generates a single k-ary de Bruijn sequence of length kn in O(1)-amortized time per bit. Based on the enumeration of both-ends-fixed k-ary necklaces of length n, we evaluate auto-correlation values of the k-ary de Bruijn sequences of length kn constructed by Sawada et al. We also estimate the asymptotic behaviour of the obtained auto-correlation values as n tends to infinity.

The top-K sorting is a variant of sorting used heavily in applications such as database management systems. Recently, the use of field programmable gate arrays (FPGAs) to accelerate sorting operation has attracted the interest of researchers. However, existing hardware top-K sorting algorithms are either resource-intensive or of low throughput. In this paper, we present a resource-efficient top-K sorting architecture that is composed of L cascading sorting units, and each sorting unit is composed of P sorting cells. K=PL largest elements are produced when a variable length input sequence is processed. This architecture can operate at a high frequency while consuming fewer resources. The experimental results show that our architecture achieved a maximum 1.2x throughput-to-resource improvement compared to previous studies.

Binary sequences with high linear complexity and high 2-adic complexity have important applications in communication and cryptography. In this paper, the 2-adic complexity of a class of balanced Whiteman generalized cyclotomic sequences which have high linear complexity is considered. Through calculating the determinant of the circulant matrix constructed by one of these sequences, the result shows that the 2-adic complexity of this class of sequences is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).

We introduce the r-ary sequence with period 2p2 derived from Euler quotients modulo 2p (p is an odd prime) where r is an odd prime divisor of (p-1). Then based on the cyclotomic theory and the theory of trace function in finite fields, we give the trace representation of the proposed sequence by determining the corresponding defining polynomial. Our results will be help for the implementation and the pseudo-random properties analysis of the sequences.

We consider all-to-all broadcast and unicast among nodes in a multi-channel single-hop ad hoc network, with no time synchronization. Motivated by the hard delay requirement for ultra-reliable and low-latency communication (URLLC) in 5G wireless networks, we aim at designing medium access control (MAC) schemes to guarantee successful node-to-node transmission within a bounded delay. To provide a hard guarantee on the transmission delay, deterministic sequence schemes are preferred to probabilistic schemes such as carrier sense multiple access (CSMA). Therefore, we mainly consider sequence schemes, with the goal to design schedule sequence set to guarantee successful broadcast/unicast within a common sequence period. This period should be as short as possible since it determines an upper bound on the transmission delay. In previous works, we have considered sequence design under time division duplex (TDD). In this paper, we focus on another common duplex mode, frequency division duplex (FDD). For the FDD case, we present a lower bound on period of feasible sequence sets, and propose a sequence construction method by which the sequence period can achieve the same order as the lower bound, for both broadcast and unicast models. We also compare the sequence length for FDD with that for TDD.

Pseudorandom sequences with low autocorrelation magnitude play important roles in various environments. Let N be a prime with N=Mf+1, where M and f are positive integers. A new method to construct M-sequences of period 4N is given. We show that these new sequences have low autocorrelation magnitude.

The direct sequence code division multiple access (DS-CDMA) technique is widely used in various communication systems. When adopting orthogonal variable spreading factor (OVSF) codes, DS-CDMA is particularly suitable for supporting multi-user/multi-rate data transmission services. A useful property of OVSF codes is that no two code sequences assigned to different users will ever interfere with each other, even if their spreading factors are different. Conventional OVSF codes are constructed based on binary orthogonal codes, called Walsh codes, and OVSF code sequences are binary sequences. In this paper, we propose new OVSF codes that are constructed based on polyphase orthogonal codes and consist of complex sequences in which each symbol is represented as a complex number. Construction of the proposed codes is based on a tree structure that is similar to conventional OVSF codes. Since the proposed codes are generalized versions of conventional OVSF codes, any conventional OVSF code can be presented as a special case of the proposed codes. Herein, we show the method used to construct the proposed OVSF codes, after which the orthogonality of the codes, including conventional OVSF codes, is investigated. Among the advantages of our proposed OVSF codes is that the spreading factor can be designed more flexibly in each layer than is possible with conventional OVSF codes. Furthermore, combination of the proposed code and a non-binary phase modulation is well suited to DS-CDMA systems where the level fluctuation of signal envelope is required to be suppressed.

Periodic sequences, used as keys in cryptosystems, plays an important role in cryptography. Such periodic sequences should possess high linear complexity to resist B-M algorithm. Sequences constructed by cyclotomic cosets have been widely studied in the past few years. In this paper, the linear complexity of n-periodic cyclotomic sequences of order 2 and 4 over 𝔽p has been calculated, where n and p are two distinct odd primes. The conclusions reveal that the presented sequences have high linear complexity in many cases, which indicates that the sequences can resist the linear attack.

Difference systems of sets (DSSs) introduced by Levenstein are combinatorial structures used to construct comma-free codes for synchronization. In this letter, two classes of optimal DSSs are presented. One class is obtained based on q-ary ideal sequences with d-form property and difference-balanced property. The other class of optimal and perfect DSSs is derived from perfect ternary sequences given by Ipatov in 1995. Compared with known constructions (Zhou, Tang, Optimal and perfect difference systems of sets from q-ary sequences with difference-balanced property, Des. Codes Cryptography, 57(2), 215-223, 2010), the proposed DSSs lead to comma-free codes with nonzero code rate.

The interval in ℕ composed of finite states of the stream version of asymmetric binary systems (ABS) is irreducible if it admits an irreducible finite-state Markov chain. We say that the stream version of ABS is irreducible if its interval is irreducible. Duda gave a necessary condition for the interval to be irreducible. For a probability vector (p,1-p), we assume that p is irrational. Then, we give a necessary and sufficient condition for the interval to be irreducible. The obtained conditions imply that, for a sufficiently small ε, if p∈(1/2,1/2+ε), then the stream version of ABS could not be practically irreducible.

Sequences that attain the smallest possible absolute sidelobes (SPASs) of periodic autocorrelation function (PACF) play fairly important roles in synchronization of communication systems, Large scale integrated circuit testing, and so on. This letter presents an approach to construct 16-QAM sequences of even periods, based on the known quaternary sequences. A relationship between the PACFs of 16-QAM and quaternary sequences is established, by which when quaternary sequences that attain the SPASs of PACF are employed, the proposed 16-QAM sequences have good PACF.

Based on the number of cyclotomy of order eight, a class of balanced almost 8-QAM sequences with odd prime periods is presented. The resultant sequences have low two-level nontrivial autocorrelation values, and their distribution is determined. Furthermore, the smallest possible absolute sidelobes (SPASs) of autocorrelation functions of balanced almost 8-QAM sequences are derived. Compared with the obtained SPASs, some of the proposed sequences is optimal or suboptimal.

Pseudo-random sequences with good statistical property, such as low autocorrelation, high linear complexity and 2-adic complexity, have been widely applied to designing reliable stream ciphers. In this paper, we explicitly determine the 2-adic complexities of two classes of generalized cyclotomic binary sequences with order 4. Our results show that the 2-adic complexities of both of the sequences attain the maximum. Thus, they are large enough to resist the attack of the rational approximation algorithm for feedback with carry shift registers. We also present some examples to illustrate the validity of the results by Magma programs.

We show that the non-trivial correlation of two properly chosen column sequences of length q-1 from the array structure of two Sidelnikov sequences of periods qe-1 and qd-1, respectively, is upper-bounded by $(2d-1)sqrt{q} + 1$, if $2leq e < d < rac{1}{2}(sqrt{q}-rac{2}{sqrt{q}}+1)$. Based on this, we propose a construction by combining properly chosen columns from arrays of size $(q-1) imes rac{q^e-1}{q-1}$ with e=2,3,...,d. The combining process enlarge the family size while maintaining the upper-bound of maximum non-trivial correlation. We also propose an algorithm for generating the sequence family based on Chinese remainder theorem. The proposed algorithm is more efficient than brute force approach.