Study On Sequences With Low Correlation And Their Related Codes

Sequences with low correlation play a very important role in cryptography, code-division multiple access (CDMA) communication system, coding, radar, sonar etc. In this thesis, we give a deep study on four topics related to sequence design, i.e., large sets of sequences with low correlation, sequences with low or zero correlation zone, optimal frequency-hopping sequences, and the application of sequences with ideal autocorrelation to error-correcting coding.Firstly, large sets of binary sequences with low correlation are investigated based on the theory of quadratic form and linearized polynomial over finite fields with even characteristic. By generalizing the construction of modified Gold sequences, a new large family of sequences with low correlation, called generalized modified Gold sequences, is obtained. The exact cor-relation values of the new sequences are calculated, and an important subclass of the new fami-lies is discussed in detail to determine its complete correlation distribution. The construction of large sets of binary sequences with low correlation and large linear complexity by Yu and Gong is extended, as a result, two new classes of large sets of binary sequences with low correlation are constructed. One class has the same period, family size, correlation, and linear complex-ity as Yu-Gong sequences. Meanwhile, the other class, compared with Yu-Gong sequences, has the the same period, correlation, and largest linear complexity, but larger family size. It is shown that the new sequence families are one of the best binary sequences known with their parameters.Large sets of p-ary sequences with low correlation are then studied using the theory of quadratic form and linearized polynomial over finite fields with odd characteristic, where p is an odd prime. By extending the constructions of Kumar-Moreno sequences with optimal cor-relation and Tan-Udaya-Fan sequences with large linear span, two new classes of large sets of p-ary sequences with low correlation are respectively proposed. The parameters of both sequence families have flexible parameters in the sense that the family size and low correla-tion can be chosen for a given sequence period, which can meet the requirement of different scenario.Secondly, the design of sequences with low or zero correlation (LCZ or ZCZ) is investi-gated based on the theory of interleaved sequences. A new uniform method for the construction of both LCZ sequence sets and ZCZ sequence sets is proposed. The resultant LCZ and ZCZ sequences sets are optimal or nearly optimal. It is the first time to construct such ZCZ sequence sets and p-ary LCZ sequence sets with flexible parameters, while for binary LCZ sequence sets, our result is better than the known result by Kim et al. It is shown that the constructions of optimal ZCZ sequence sets by Matsufuji et al and Hayashi et al are essentially interleaved con-structions and have the same peculiarity that the resulting ZCZ sequences are cyclically equiv-alent. Addressing this problem, a new general construction of ZCZ sequences is presented. By the new method, a new class of optimal ZCZ sequences which are pairwise cyclically distinct is constructed. By combining the theory of interleaved sequences and quadratic form, new approximately optimal binary LCZ sequences are constructed.Next, the zero-difference balanced (ZDB) functions recently proposed by Ding are stud-ied. ZDB functions, as a generalization of perfect nonlinear (PN) functions, have important applications in coding design. Based on the concept of partitioned difference families, the combinatorial characteristic of ZDB functions is given and an intimate connection between ZDB functions and the partitioned difference family is established. Three new classes of ZDB functions with flexible parameters are constructed using difference-balanced functions which are originated from sequences with ideal autocorrelation. The ZDB functions constructed by Ding from trace function and ternary sequences with ideal autocorrelation are just special cases of the new ones.A direct bridge from ZDB functions and frequency-hopping sequences (FHSs) is estab-lished. Using the connection, three new general constructions of optimal FHS sets are proposed in terms of the new ZDB functions, which generate three new classes of optimal FHS sets. The first construction unifies the classical construction of optimal FHS sets based on m-sequences over finite fields and finite rings, the second one contains the constructions based on the deci-mation of m-sequences as special cases, while the third one could produce FHS sets with more flexible parameters than the construction by Chung et al. It is shown that FHSs with large complexity can be constructed from the new constructions.Finally, using the new ZDB functions, new optimal constant-composition codes (CCCs) with flexible parameters, new optimal constant-weight codes (CWCs) with flexible parameters, and new optimal difference systems of sets (DSSs) with flexible parameters, are obtained. Some known CCCs and DSSs are just special cases of the new results.