Research On Frequency Hopping Sequences And Sequences With Low Correlation

Pseudo-random sequences have been playing an important role for several decades in modern communication systems such as code division multiple access (CDMA), code, radar, sonar and cryptography. In particular, there are hot topics in both theory and application for pseudorandom sequences such as frequency hopping sequences, p-ary m sequences with low correlation and sequences with large linear complexity. These systems promise all users to use a common frequency/time band simultaneously in order to improve the utilization of the band and every sequence represents a user. The sequence with low correlation reduces collisions which are caused by simultaneous transmission of a user or the others.The main aim of this work was devoted to the design of frequency hopping se-quence(FHS) with nice correlation property, the design of p-ary sequences with low correlation, and the calculation of the linear complexity of a new binary sequence.Firstly, in design of FHS with nice correlation property, for the periodic Hamming correlation, if any two primes p and q satisfy some conditions, by using the generalized cyclotomy and the Chinese Remainder Theorem, we construct a new class of optimal frequency hopping sequence set with flexible parameters with respect to the Peng-Fan bound. Furthermore, each FHS of the set is near optimal with respect to the Lempel-Greenberger bound. This set has large size, large period of these sequences which generalizes the early results given by Ding and Zhang based on the similar technique. Meanwhile, for the average Hamming correlation, given any window length at most the period of FHSs, there are no corresponding results on an FHS set with APHC so far. Thus we propose first a low bound of an FHS set with APHC for the given window length. And we also give a sufficient and necessary condition when an FHS set sat-isfies the optimal average partial Hamming correlation which can be seen as a bridge between optimal average partial Hamming correlation and optimal average Hamming correlation. We further construct an FHS set with optimal average partial Hamming correlation by feat of the Chinese Remainder Theorem and the generalized cyclotomy. Secondly, for the purpose of design of a class of p-ary sequence set with low cor-relation, by applying quadratic form and linearized polynomial, we discuss the upper bound of cross correlation between a class of p-ary m sequence and its decimated se-quence by a new decimated value. This result generalizes the case of the prime3given by Kim to a general case of any prime p which is congruent to3modulo4. In addition, we construct a p-ary sequence set with big size and low correlation.Finally, in view of the requirement of large linear complexity of a sequence, for any distinct two odd primes p and q, we propose a new class of sequence of period which is the product of two distinct odd primes using two fold cyclotomy and compute its linear complexity and minimal polynomial.