Applications Of Combination In Cyclic Codes And Sequences

By using combinatorial method to investigate cyclic codes and sequences,the thesis mainly computes the weight distributions of cyclic codes,determines Whiteman's generalized cyclotomic numbers,and constructs cyclotomic sequences with good cryptographic properties.The weight distributions of cyclic codes are very important in the research of coding and decoding theory.Firstly,the thesis investigates the weight distributions of cyclic codes of length ml over a finite field qF,where q is a power of a prime and l is a prime satisfying lv||(q-1).By the methods of combination,exponential sums and solutions of systems of equations and so on,the thesis gives the weight distributions of irreducible codes and reducible codes whose dual codes have two or three zeros,as well as computes the complete weight distributions of irreducible cyclic codes.Secondly,the thesis investigates the weight distributions of the cyclic codes of length mtl,where q is a positive power of a prime and l is a prime satisfying ||(1)vl q-,l|/t and t|(q-1).The thesis adopts combinatorial method to compute the solution number of a special system of equations over the finite field,works out the weight distribution of the cyclic code whose dual code has arbitrary zeros,which answers an open problem asked by Dr.Jing Yang et al.,and obtains the weight distributions of some cyclic codes of length 2ml,4ml and 8ml if q ?3(mod 4).Thirdly,the thesis investigates Whiteman's generalized cyclotomic number of order four.Whiteman's generalized cyclotomic is widely used in pseudorandom sequences with good cryptographic properties.Let p,q be distinct primes with gcd(p-1,q-1)=4,Whiteman gave the sixteen cyclotomic numbers depending solely upon one of the two decompositions:pq =a +4b,pq =a?2 +4b?2,a ?a? ?1(mod 4).This thesis independently determines the sixteen cyclotomic numbers of order four depending on unique a,b if a common primitive root of p and q is chosen.Finally,the thesis constructs a nearly balanced binary Whiteman's generalized cyclotomic sequence such that it has good autocorrelation and high linear complexity under certain conditions.The thesis computes the reciprocal minimal polynomial of the sequence,constructs a cyclic code such that the reciprocal minimal polynomial is its parity-check polynomials,and determines the weight distribution of the cyclic code.