Study On Properties And Applications Of Pseudo-random Sequences

Pseudo-random sequences having good autocorrelation and large linear complexityare found wide applications in many engineering felds, such as spread spectrum communi-cation，cryptography and coding technology etc. First, this dissertation presents severaldecimation values of m-sequences. For each decimation value d, the distribution or theupper bound of the cross-correlation function between an m-sequence and its decimatedsequence by the decimation value d is determined. Second, the linear complexity of afamily of sequences with three-level autocorrelation is computed in this dissertation, anda generalization of a known construction of sequence families with three-level autocor-relation is given. Finally, several classes of cyclic codes are constructed using the frstclass Whiteman generalized cyclotomic sequence of order4, and the corresponding lowerbound of the minimum distance of the cyclic code is also given.The author obtains main results as follows:（1）. Let p be an odd prime satisfying p≡1mod4, n a positive integer satisfyingn=2k, k an odd interger, e|k. For an m-sequence s of period pⁿ1and adecimation value d=（p^k+1）2/2（p^e+1）, we prove that the cross correlation function between anm-sequence and its decimation s^（d）is6-valued, and we also compute the distributionof the cross-correlation values. The result shows that the magnitude of the crosscorrelation value is upper bounded by2√pⁿ+1, which is very meaningful in theCDMA communication system.（2）. Let p be an odd prime, n a positive integer and n=2m. For an m-sequence s ofperiod pⁿ1and a decimation value of Niho type d=（p^m1）²/2+1, we prove that thecross-correlation function between the sequence s and its decimation sequence s^（d）is6-valued. The result shows that for any p, n, m and d satisfying the above conditionthe magnitude of the cross-correlation function is upper bounded by4√pⁿ1.（3）. Ness and Helleseth initiated studies on the cross-correlation between two binary m-sequences of diferent periods. Based on their results, we analyze the properties ofthe cross-correlation between two p-ary m-sequences of diferent periods, and derivethe equations for the cross-correlation values to be met. Finally, for p=3, it isproved that the cross-correlation values takes on at least three diferent values.（4）. In2009, employing a class of cyclic diference sets with Singer parameters and aclass of cyclic relative diference sets, Cai and Ding constructed a class of cyclicalmost diference sets which corresponds to a class of binary sequences with3-level autocorrelation. For one family of sequences, i.e., the family based on the WG diference sets, of those constructed by Cai and Ding, we derive its minimalpolynomial and compute its linear complexity.（5）. We present a generalization of the construction by Cai and Ding, i.e. using d-formfunctions with diference balance properties instead of the simple trace function togenerate a class of more general cyclic relative diference sets than the one employedby Cai and Ding, we obtain more families of sequences with3-level autocorrelationthan those constructed by Cai and Ding.（6）. Using Whiteman’s cyclotomic sequences of order4and prime period, Ding con-structed several classes of q-ary cyclic codes. Inspired by the classifcation methodadopted by Ding, we investigate several classes of q-ary cyclic codes generated by thefrst class Whiteman’s generalized cyclotomic sequences of order4and two-primeperiod based on the properties of the frst class whiteman’s generalized cyclotomicsequence of order4. And we obtain their lower bounds on the minimal distances ofthese cyclic codes. Our results show that the lower bounds on the minimal distancesof these cyclic codes is equal to the corresponding lower bounds on the minimal dis-tances of those cyclic codes constructed by Ding.