Linear Complexity and Trace Representation of New Ding Generalized Cyclotomic Sequences with Period pq and Order Two

Linear complexity is an important property to measure the unpredictability of pseudo-random sequences. Trace representation is helpful for analyzing cryptography properties of pseudo-random sequences. In this paper, a class of new Ding generalized cyclotomic binary sequences of order two with period pq is constructed based on the new segmentation of Ding Helleseth generalized cyclotomy. Firstly, the linear complexity and minimal polynomial of the sequences are investigated. Then, their trace representation is given. It is proved that the sequences have larger linear complexity and can resist the attack of the Berlekamp–Massey algorithm. This paper also confirms that generalized cyclotomic sequences with good randomness may be obtained by modifying the characteristic set of generalized cyclotomy.

Analysis of linear complexity of generalized cyclotomic Q-ary sequences of pn period

The article presents the analysis of the linear complexity of periodic q-ary sequences when changing k of their terms per period. Sequences are formed on the basis of new generalized cyclotomy modulo equal to the degree of an odd prime. There has been obtained a recurrence relation and an estimate of the change in the linear complexity of these sequences, where q is a primitive root modulo equal to the period of the sequence. It can be inferred from the results that the linear complexity of these sequences does not sign ificantly decrease when k is less than half the period. The study summarizes the results for the binary case obtained earlier.

Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two

Suppose that p and q are two distinct odd prime numbers with n = p q . In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over ℤ n was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over F l was presented and their minimal polynomials and linear complexities were derived, where l = r s with a prime number r and gcd l , n = 1 . The results have indicated that the linear complexities of these sequences are high without any special requirements on the prime numbers. Furthermore, we employed these sequences to obtain a few cyclic codes over F l with length n and developed the lower bounds of the minimum distances of many cyclic codes. It is important to stress that some cyclic codes in this paper are optimal.

Linear Complexity of a New Class of Quaternary Generalized Cyclotomic Sequence with Period 2pm

Sequences with high linear complexity property are of importance in applications. In this paper, based on the theory of generalized cyclotomy, new classes of quaternary generalized cyclotomic sequences with order 4 and period 2pm are constructed. In addition, we determine their linear complexities over finite field F4 and over ℤ4, respectively.

Cyclotomy and arithmetic properties of some families in Z[x]/〈xpq − 1〉

Cyclotomic classes of order 2 with respect to a product of two distinct odd primes [Formula: see text] and [Formula: see text] are represented in some specific forms and using these forms an alternate proof of Theorem 3 of [C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl. 4 (1998) 140–166] is given, when [Formula: see text]. Further, it is observed that these classes are related to [Formula: see text]-cyclotomic cosets, where [Formula: see text] and [Formula: see text] such that gcd([Formula: see text]. Finally, arithmetic properties of some families in [Formula: see text] and hence in [Formula: see text] are studied.

Optimal Frequency-Hopping Sequence Sets Based on Cyclotomy

In this paper, a kind of generalized cyclotomy with respect to the square of a prime is presented and the properties of the corresponding generalized cyclotomic numbers are investigated. Based on the generalized cyclotomy, a class of frequency-hopping sequence (FHS) set is constructed. By means of some basic properties of the generalized cyclotomy, we derive the Hamming correlation distribution of the new set. The results show that the proposed FHS set is optimal with regard to the average Hamming correlation (AHC) bound. By choosing suitable parameters, the construction also leads to the optimal FHS set and the optimal FHSs with regard to the maximum Hamming correlation (MHC) bound and Lempel-Greenberger bound, respectively.

NEW OPTIMAL FREQUENCY HOPPING SEQUENCE SETS FROM BALANCED NESTED DIFFERENCE PACKINGS OF PARTITION-TYPE

In this paper, we construct a new kind of balanced nested difference packings of partition-type based on generalized cyclotomy with respect to ℤpm. As an application, we propose a construction of frequency hopping sequence sets. The frequency hopping sequence sets obtained in this paper are optimal with respect to the Peng-Fan bound.