Recently, trace representation of a class of balanced quaternary sequences of period p from the classical cyclotomic classes was given by Yang et al. (Cryptogr. Commun.,15 (2023): 921-940). In this letter, based on the generalized cyclotomic classes, we define a class of balanced quaternary sequences of period pn, where p = ef + 1 is an odd prime number and satisfies e ≡ 0 (mod 4). Furthermore, we calculate the defining polynomial of these sequences and obtain the formula for determining their trace representations over ℤ4, by which the linear complexity of these sequences over ℤ4 can be determined.

A unified construction for yielding optimal and balanced quaternary sequences from ideal/optimal balanced binary sequences was proposed by Zeng et al. In this paper, the linear complexity over finite field 𝔽2, 𝔽4 and Galois ring ℤ4 of the quaternary sequences are discussed, respectively. The exact values of linear complexity of sequences obtained by Legendre sequence pair, twin-prime sequence pair and Hall's sextic sequence pair are derived.

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.

In this letter, two new families of quaternary sequences with low four-level or five-level autocorrelation are constructed based on generalized cyclotomy over Z2p. These quaternary sequences are balanced and the maximal absolute value of the out-of-phase autocorrelation is 4.

A family of quaternary sequences over Z4 is defined based on the Ding-Helleseth generalized cyclotomic classes modulo pq for two distinct odd primes p and q. The linear complexity is determined by computing the defining polynomial of the sequences, which is in fact connected with the discrete Fourier transform of the sequences. The results show that the sequences possess large linear complexity and are “good” sequences from the viewpoint of cryptography.

Let r be an odd prime, such that r≥5 and r≠p, m be the order of r modulo p. Then, there exists a 2pth root of unity in the extension field Frm. Let G(x) be the generating polynomial of the considered quaternary sequences over Fq[x] with q=rm. By explicitly computing the number of zeros of the generating polynomial G(x) over Frm, we can determine the degree of the minimal polynomial, of the quaternary sequences which in turn represents the linear complexity. In this paper, we show that the minimal value of the linear complexity is equal to $ rac{1}{2}(3p-1) $ which is more than p, the half of the period 2p. According to Berlekamp-Massey algorithm, these sequences viewed as enough good for the use in cryptography.

Pseudo-random sequences with high linear complexity play important roles in many domains. We give linear complexity of generalized cyclotomic quaternary sequences with period pq over Z4 via the weights of its Fourier spectral sequence. The results show that such sequences have high linear complexity.

In this paper, one new class of quaternary generalized cyclotomic sequences with the period 2pq over F4 is established. The linear complexity of proposed sequences with the period 2pq is determined. The results show that such sequences have high linear complexity.

In this paper, for an odd prime p, new quaternary sequences of even period 2p with ideal autocorrelation property are constructed using the binary Legendre sequences of period p. For the new quaternary sequences, two properties which are considered as the major characteristics of pseudo-random sequences are derived. Firstly, the autocorrelation distribution of the proposed quaternary sequences is derived and it is shown that the autocorrelation values of the proposed quaternary sequences are optimal. For both p≡1 mod 4 and p≡3 mod 4, we can construct optimal quaternary sequences while only for p≡3 mod 4, the binary Legendre sequences can satisfy ideal autocorrelation property. Secondly, the linear complexity of the proposed quaternary sequences is also derived by counting non-zero coefficients of the discrete Fourier transform over the finite field Fq which is the splitting field of x2p-1. It is shown that the linear complexity of the quaternary sequences is larger than or equal to p or (3p+1)/2 for p≡1 mod 4 or p≡3 mod 4, respectively.

A unified construction for transforming binary sequences of balance or unbalance into quaternary sequences is presented. On the one hand, when optimal and balanced binary sequences with even period are employed, our construction is exactly the same Jang, et al.'s and Chung, et al.'s ones, which result in balanced quaternary sequences with optimal autocorrelation magnitude. On the other hand, when ideal and balanced binary sequences with odd period N are made use of, our construction produces new balanced quaternary sequences with optimal autocorrelation value (OAV), in which there are N distinct sequences in terms of cyclic shift equivalence, and includes Tang, et al.'s and Jang, et al.'s ones as special cases. In addition, when binary sequences without period 2n-1 or balance are employed, the transformation of Jang, et al.'s method is invalid, however, the proposed construction works very good. As a consequence, this unified construction allows us to construct optimal and balanced quaternary sequences from ideal/optimal balanced binary sequences with arbitrary period.

The linear complexity of quaternary sequences plays an important role in cryptology. In this paper, the minimal polynomial of a class of quaternary sequences with low autocorrelation constructed by generalized cyclotomic sequences pairs is determined, and the linear complexity of the sequences is also obtained.

This letter presents a framework, including two constructions, for yielding several types of sequences with optimal autocorrelation properties. Only by simply choosing proper coefficients in constructions and optimal known sequences, two constructions transform the chosen sequences into optimally required ones with two or four times periods as long as the original sequences', respectively. These two constructions result in binary and quaternary sequences with optimal autocorrelation values (OAVs), perfect QPSK+ sequences, and multilevel perfect sequences, depending on choices of the known sequences employed. In addition, Construction 2 is a generalization of Construction B in [5] so that the number of distinct sequences from the former is larger than the one from the latter.

In contemporary communications, Golay, periodic and Z- complementary sequence sets play a very important role, since such sequence sets possess impulse-like or zero correlation zone (ZCZ) autocorrelation. On the other hand, the advantages of the signals over the quadrature amplitude modulation (QAM) constellation are more and more prominent. Hence, the design of such sequence sets over the QAM constellation has turned into one of the all-important issues in communications. Therefore, the construction methods of such sequence sets over the 16-QAM constellation are investigated, in this letter, and our goals are arrived at by the known quaternary Golay, periodic and Z- complementary sequence sets. Finally, many examples illuminate the validity of the proposed methods.

We determine the autocorrelations of the quaternary sequence over F4 and its modified version introduced by Du et al. [X.N. Du et al., Linear complexity of quaternary sequences generated using generalized cyclotomic classes modulo 2p, IEICE Trans. Fundamentals, vol.E94-A, no.5, pp.1214–1217, 2011]. Furthermore, we reveal a drawback in the paper aforementioned and remark that the proof in the paper by Kim et al. can be simplified.

In this correspondence, a new method to extend the number of quaternary low correlation zone (LCZ) sequence sets is presented. Based on the inverse Gray mapping and a binary sequence with ideal two-level auto-correlation function, numbers of quaternary LCZ sequence sets can be generated by choosing different parameters. There is at most one sequence cyclically equivalent in different LCZ sequence sets. The parameters of LCZ sequence sets are flexible.

In this letter, we propose a new construction of quaternary low correlation zone (LCZ) sequence set using binary LCZ sequence sets and an inverse Gray mapping. The new construction method provides optimal quaternary LCZ sequence sets even if the employed binary LCZ sequence set is suboptimal. The optimality is improved at the price of alphabet extension.

In this paper, a new construction method of quaternary sequences of even period 2N having the ideal autocorrelation and balance properties is proposed. These quaternary sequences are constructed by applying the inverse Gray mapping to binary sequences of odd period N with the ideal autocorrelation. Autocorrelation distribution of the proposed quaternary sequences is derived. These sequences can be used to construct quaternary sequence families of even period 2N. Family size and the maximum absolute value of correlation spectrum of the proposed quaternary sequence families are also derived.

Let p be an odd prime number. We define a family of quaternary sequences of period 2p using generalized cyclotomic classes over the residue class ring modulo 2p. We compute exact values of the linear complexity, which are larger than half of the period. Such sequences are 'good' enough from the viewpoint of linear complexity.

We propose an extension method of quaternary low correlation zone (LCZ) sequence set with odd period. From a quaternary LCZ sequence set with parameters (N, M, L, 1), the proposed method constructs a new quaternary LCZ sequence set with parameters (2N, 2M, L, 2), where N is odd. If the employed LCZ sequence set in the construction is optimal, the extended LCZ sequence set becomes also optimal where N = kL, L > 4, and k>2.

In this paper we present a construction method for quaternary sequences from a binary sequence of even period, which preserves the period and autocorrelation of the given binary sequence. By applying the method to the binary sequences with three-valued autocorrelation, we construct new quaternary sequences with three-valued autocorrelation, which are balanced or almost balanced. In particular, we construct new balanced quaternary sequences whose autocorrelations are three-valued and have out-of-phase magnitude 2, when their periods are N=pm-1 and N≡ 2 (mod 4) for any odd prime p and any odd integer m. Their out-of-phase autocorrelation magnitude is the known optimal value for N≠ 2,4,8, and 16.