Cross Correlation Of Sequences And Constructions And Weight Distributions Of Cyclic Codes

Periodic sequences are closely related to codewords of cyclic codes,there is a one-to-one correspondence between periodic sequences and equivalent classes of codewords of cyclic codes under the equivalence of cyclic shift.With this connection,a sequence set can be constructed from a class of cyclic codes and vice versa.The cross correlation function between sequences is one of important indexes for characterizations of pseudorandom sequences.The weight distribution of cyclic codes not only reflects the error correction ability,but also calculates the error probability of error detection and error correction for some error detection and error correction algorithm.Based on the expression of trace function of sequences and cyclic codes,the correlation function between sequences and the weight of each codeword of cyclic codes can be formulated by exponential sums.So the correlation distribution of sequences and the weight distribution of cyclic codes can be determined by means of the value distribution of exponential sums.Utilizing exponential sums and solutions of equations over finite fields,we investigate cross correlation distributions of m-sequences and their certain decimation sequences,and constructions and weight distributions of cyclic codes.Sequences with low auto or cross correlation have important applications in cryptography and CDMA communication systems.The maximum length linear sequences(or m-sequences)and their decimations are widely used to design families of sequences with low auto or cross correlation.It is interesting to find new decimaions such that the cross correlation functions between an m-sequence and its decimation sequences have a few correlation values.A new class of Niho exponents d resulting in four-valued cross correlation is firstly found in this thesis.By the polar coordinate representations of elements in the finite field,the value distribution of a class of exponential sums corresponding to d is completely determined.When the largest common divisor of d and the period of an m-sequence is equal to 3,there are three distinct d-decimation sequences.The cross correlation distribution of the m-sequence and each of its d-decimation sequences is determined by means of the value distribution of the corresponding exponential sums.Moreover,the weight distribution of a class of cyclic codes related to d having two nonzeros and four weights is also determined.And the correlation distribution among sequences in a family of sequences is presented,some of them are low correlation.Ideal correlation sequences constructed by multiphase p-ary sequences can satisfy the requirement of increasing address codes in CDMA communication systems to a certain extent.A class of odd prime p-ary Niho decimations is found.It is proved that the cross correlation function between a p-ary m-sequence and the summation of these decimation sequences is three-valued or four-valued,and the distribution is also determined by means of exponential sums and solutions of certain equations over finite fields.Also,based on solutions of certain equations and properties of perfect nonlinear functions and almost perfect nonlinear functions over finite fields,the necessary and sufficient condition for the quinary cyclic codes with generator polynomial(x + 1)m?(x)m?e(x)to have parameters[5m-1,5m-2m-2,4]is provided by an-alyzing irreducible factors of certain polynomials over finite fields.And thus several classes of new optimal quinary cyclic codes with the same parameters and generator polynomial are constructed.