This paper focuses on a pseudorandom number generator called an NTU sequence for use in cryptography. The generator is defined with an m-sequence and Legendre symbol over an odd characteristic field. Since the previous researches have shown that the generator has maximum complexity; however, its bit distribution property is not balanced. To address this drawback, the author introduces dynamic mapping for the generation process and evaluates the period and some distribution properties in this paper.

The shift-and-add property (SAP) of a p-ary m-sequence {ak} with period N=pn-1 means that this sequence satisfies the equation {ak+η}+{ak+τ}={ak+λ} for some integers η, τ and λ. For an arbitrarily-given p-ary m-sequence {ak}, we develop an algebraic approach to determine the integer λ for the arbitrarily-given integers η and τ. And all trinomials can be given. Our calculation only depends on the reciprocal polynomial of the primitive polynomial which produces the given m-sequence {ak}, and the cyclotomic cosets mod pn-1.

Let d=2pm-1 be the Niho decimation over $mathbb{F}_{p^{2m}}$ satisfying $gcd(d,p^{2m}-1)=3$, where m is an odd positive integer and p is a prime with p ≡ 2(mod 3). The cross-correlation function between the p-ary m-sequence of period p2m-1 and its every d-decimation sequence with short period $rac{p^{2m}-1}{3}$ is investigated. It is proved that for each d-decimation sequence, the cross-correlation function takes four values and the corresponding correlation distribution is completely determined. This extends the results of Niho and Helleseth for the case gcd(d, p2m-1)=1.

For the reconstruction of the feedback polynomial of a synchronous scrambler placed after a convolutional encoder, the existing algorithms require the prior knowledge of a dual word of the convolutional code. To address the case of a dual word being unknown, a new algorithm for the reconstruction of the feedback polynomial based on triple correlation characteristic of an m-sequence is proposed. First, the scrambled convolutional code sequence is divided into bit blocks; the product of the scrambled bit blocks with a dual word is proven to be an m-sequence with the same period as the synchronous scrambler. Second, based on the triple correlation characteristic of the generated m-sequence, a dual word is estimated; the generator polynomial of the generated m-sequence is computed by two locations of the triple correlation peaks. Finally, the feedback polynomial is reconstructed using the generator polynomial of the generated m-sequence. As the received sequence may contain bit errors, a method for detecting triple correlation peaks based on the constant false-alarm criterion is elaborated. Experimental results show that the proposed algorithm is effective. Ulike the existing algorithms available, there is no need to know a dual word a priori and the reconstruction result is more accurate. Moreover, the proposed algorithm is robust to bit errors.

In this paper, for an odd prime p, two positive integers n, m with n=2m, and pm≡1 (mod 4), we derive an upper bound on the magnitude of the cross-correlation function between two decimated sequences of a p-ary m-sequence. The two decimation factors are 2 and 2(pm+1), and the upper bound is derived as $rac{3}{2}p^m + rac{1}{2}$. In fact, those two sequences correspond to the p-ary sequences used for the construction of Kasami sequences decimated by 2. This result is also used to obtain an upper bound on the cross-correlation magnitude between a p-ary m-sequence and its decimated sequence with the decimation factor $d=rac{(p^m +1)^2}{2}$.

Based on the work by Helleseth [1], for an odd prime p and an even integer n=2m, the cross-correlation values between two decimated m-sequences by the decimation factors 2 and 4pn/2-2 are derived. Their cross-correlation function is at most 4-valued, that is, $igg {rac{-1 pm p^{n/2}}{2}, rac{-1 + 3p^{n/2}}{2}, rac{-1 + 5p^{n/2}}{2} igg }$. From this result, for pm ≠ 2 mod 3, a new sequence family with family size 4N and the maximum correlation magnitude upper bounded by $rac{-1 + 5p^{n/2}}{2} simeq rac{5}{sqrt{2}}sqrt{N}$ is constructed, where $N = rac{p^n-1}{2}$ is the period of sequences in the family.

Let p be an odd prime such that p ≡ 3 mod 4 and n be an odd positive integer. In this paper, two new families of p-ary sequences of period $N = rac{p^n-1}{2}$ are constructed by two decimated p-ary m-sequences m(2t) and m(dt), where d=4 and d=(pn+1)/2=N+1. The upper bound on the magnitude of correlation values of two sequences in the family is derived by using Weil bound. Their upper bound is derived as $rac{3}{sqrt{2}} sqrt{N+rac{1}{2}}+rac{1}{2}$ and the family size is 4N, which is four times the period of the sequence.

Let p be an odd prime and m be any positive integer. Assume that n=2m and e is a positive divisor of m with m/e being odd. For the decimation factor $d=rac{(p^{m}+1)^2}{2(p^e+1)}$, the cross-correlation between the p-ary m-sequence ${tr_1^n(alpha^i)}$ and its decimated sequence ${tr_1^n(alpha^{di})}$ is investigated. The value distribution of the correlation function is completely determined. The results in this paper generalize the previous results given by Choi, Luo and Sun et al., where they considered some special cases of the decimation factor d with a restriction that m is odd. Note that the integer m in this paper can be even or odd. Thus, the decimation factor d here is more flexible than the previous ones. Moreover, our method for determining the value distribution of the correlation function is different from those adopted by Luo and Sun et al. in that we do not need to calculate the third power sum of the correlation function, which is usually difficult to handle.

Let m ≥ 3 be an odd positive integer, n=2m and p be an odd prime. For the decimation factor $d=rac{(p^{m}+1)(p^{m}+p-1)}{p+1}$, the cross-correlation between the p-ary m-sequence {tr1n(αt)} and its all decimated sequences {tr1n(αdt+l)} is investigated, where 0 ≤ l < gcd(d,pn-1) and α is a primitive element of Fpn. It is shown that the cross-correlation function takes values in {-1,-1±ipm|i=1,2,…p}. The result presented in this paper settles a conjecture proposed by Kim et al. in the 2012 IEEE International Symposium on Information Theory Proceedings paper (pp.1014-1018), and also improves their result.

In this paper, for an odd prime p such that p≡3 mod 4, odd n, and d=(pn+1)/(pk+1)+(pn-1)/2 with k|n, the value distribution of the exponential sum S(a,b) is calculated as a and b run through $mathbb{F}_{p^n}$. The sequence family $mathcal{G}$ in which each sequence has the period of N=pn-1 is also constructed. The family size of $mathcal{G}$ is pn and the correlation magnitude is roughly upper bounded by $(p^k+1)sqrt{N}/2$. The weight distribution of the relevant cyclic code C over $mathbb{F}_p$ with the length N and the dimension ${ m dim}_{mathbb{F}_p}mathcal{C}=2n$ is also derived.

In this paper, we analyze the existing results to derive the cross-correlation distributions of p-ary m-sequences and their decimated sequences for an odd prime p and various decimations d. Based on the previously known results, a methodology to obtain the distribution of their cross-correlation values is also formulated.

In this paper, N-CSK (N parallel Codes Shift Keying) using modified pseudo orthogonal M-sequence sets (MPOMSs) to realize the parallel combinatory spread spectrum (PC/SS) communication system for the optical communications is proposed. Moreover, the upper bound of data transmission rate and the bit error rate (BER) performance of this N-CSK system using the chip-level detection are evaluated through theoretical analysis by taking into account the scintillation, background-noise, avalanche photo-diode (APD) noise, thermal noise, and signal dependence noise. It is shown that the upper bound of data transmission rate of the proposed system is better than those of OOK/CDM and SIK/CDM. Moreover, the upper bound of data transmission rate of the proposed system can achieve about 1.5 [bit/chip] when the code length of MPOMS is 64 [chip].

This paper presents and evaluates a new acoustic imaging system that uses multicarrier signals for correlation division in synthetic transmit aperture (CD-STA). CD-STA is a method that transmits uncorrelated signals from different transducers simultaneously to achieve high-speed and high-resolution acoustic imaging. In CD-STA, autocorrelations and cross-correlations in transmitted signals must be suppressed because they cause artifacts in the resulting images, which narrow the dynamic range as a consequence. To suppress the correlation noise, we had proposed to use multicarrier signals optimized by a genetic algorithm. Because the evaluation of our proposed method was very limited in the previous reports, we analyzed it more deeply in this paper. We optimized three pairs of multicarrier waveforms of various lengths, which correspond to 5th-, 6th- and 7th-order M-sequence signals, respectively. We built a CD-STA imaging system that operates in air. Using the system, we conducted imaging experiments to evaluate the image quality and resolution of the multicarrier signals. We also investigated the ability of the proposed method to resolve both positions and velocities of target scatterers. For that purpose, we carried out an experiment, in which both moving and fixed targets were visualized by our system. As a result of the experiments, we confirmed that the multicarrier signals have lower artifact levels, better axial resolution, and greater tolerance to velocity mismatch than M-sequence signals, particularly for short signals.

Binary maximal-length sequences (or m-sequences) are sequences of period 2m-1 generated by a linear recursion of degree m. Decimating an m-sequence {st} by an integer d relatively prime to 2m-1 leads to another m-sequence {sdt} of the same period. The crosscorrelation of m-sequences has many applications in communication systems and has been an important and well studied problem during more than 40 years. This paper presents an updated survey on the crosscorrelation between binary m-sequences with at most five-valued crosscorrelation and shows some of the many recent connections of this problem to several areas of mathematics such as exponential sums and Dickson polynomials.

In this paper, the cross-correlation distribution between a p-ary m-sequence s(t) and its p + 1 distinct decimated sequences s(dt + l) is derived. For an odd prime p, an even integer n, and d = pk +1 with gcd(n, k) = 1, there are p + 1 distinct decimated sequences s(dt + l), 0 ≤ l < p + 1, for a p-ary m-sequence s(t) of period pn -1 because gcd(d, pn - 1) = p + 1. The maximum magnitude of their cross-correlation values is 1 + p if l ≡ 0 mod p + 1 for n ≡ 0 mod 4 or l ≡ (p + 1)/2 mod p + 1 for n ≡ 2 mod 4 and otherwise, 1 + . Also by using s(t) and s(dt + l), a new family of p-ary sequences of period pn -1 is constructed, whose family size is pn and Cmax is 1 + p.

Convolutional spreading CDMA with cyclic prefix (CS-CDMA/CP) is a multiuser interference-free (MUI-free) CDMA scheme proposed for multipath channels based on the convolution between user data and zero correlation zone (ZCZ) code, and its characteristics depend on the employed ZCZ codes. Although ternary ZCZ codes have more sequences than binary ZCZ codes in general, transmitted signal with ternary ZCZ codes give a slightly higher peak-to-average power ratio (PAPR). In this paper we propose the use of periodic ZCZ codes generated from an M-sequence which not only provides the same user capacity as ternary ZCZ codes but allows more design flexibility. Simulation results show that the new ZCZ code shows stronger robustness against an imperfect transmitter with clipping and enjoys better BER performances when used in CS-CDMA/CP compared to the conventional DS-CDMA with MRC-RAKE.

Optical fiber communications require multiple-access schemes to access a shared channel among multiple users. The coherent ultrashort light pulse code-division multiple-access (CDMA) system is one such scheme, and it also offers asynchronous-access communication. This system usually employs 2-level, i.e., binary, m-sequences as signature codes because of their low correlation. If the number of active users is greater than the length of the m-sequence, i.e., code length, distinct m-sequences are used. However, the distinct 2-level m-sequences do not exhibit low correlation, resulting in performance degradation. We therefore propose a coherent ultrashort light pulse CDMA communication system with distinct 4-level, i.e., quaternary, m-sequences to improve system performance when the number of users is greater than the code length. We created the 4-level m-sequences from 2-level m-sequences, and assess the correlation of the 4-level m-sequences. We also theoretically derive the bit error rate (BER) of the proposed system taking into account multiple-access interference (MAI), beat noise, amplified spontaneous emission (ASE), shot noise, and thermal noise. The numerical results show that BER for distinct 4-level m-sequences is more than an order of magnitude smaller than that for distinct 2-level m-sequences. BER is limited by MAI and beat noise when the power of the received signal is high, otherwise BER is limited by ASE, shot noise, and thermal noise.

Binary sequences with two-level periodic autocorrelation correspond directly to cyclic (v, k, λ)-designs. When v = 4t-1, k = 2t -1 and λ = t-1, for some positive integer t, the sequence (or design) is called a cyclic Hadamard sequence (or design). For all known examples, v is either a prime number, a product of twin primes, or one less than a power of 2. Except when v = 2k-1, all known examples are based on quadratic residues (using the Legendre symbol when v is prime, and the Jacobi symbol when v = p(p+2) where both p and p+2 are prime); or sextic residues (when v is a prime of the form 4a2 + 27). However, when v = 2k-1, many constructions are now known, including m-sequences (corresponding to Singer difference sets), quadratic and sextic residue sequences (when 2k-1 is prime), GMW sequences and their generalizations (when k is composite), certain term-by-term sums of three and of five m-sequences and more general sums of trace terms, several constructions based on hyper-ovals in finite geometries (found by Segre, by Glynn, and by Maschietti), and the result of performing the Welch-Gong transformation on some of the foregoing.

The present paper introduces an improved construction of a class of binary sequences having a zero-correlation zone (hereafter binary ZCZ sequence set). The cross-correlation function and the side-lobe of the auto-correlation function of the proposed sequence set is zero for the phase shifts within the zero-correlation zone. The present paper shows that such a construction generates a binary ZCZ sequence set from an arbitrary M-sequence. The previously reported sequence construction of binary ZCZ sequence sets from an M-sequence can generate a single series of binary ZCZ sequence sets from an M-sequence. The present paper proposes an improved sequence construction that can generate more than one series of binary ZCZ sequence sets from an M-sequence.

In the letter, properties of m-sequence are derived, based on these properties, a family of binary nonlinear constant weight codes is presented, these binary nonlinear constant weight codes can apply to automatic repeat request (ARQ) communication system, as detecting-error codes.