Research On Several Problems Of Linear Error-correcting Codes

Exponential sums over finite fields not only are fundamental objects in number theory, but also have wide applications in coding and cryptography. The thesis uses exponential sums over finite fields to determine the linear complexity and the minimal polynomials of cyclotomic sequences, as well as the weight distributions of some classes of cyclic codes. Meanwhile, this dissertation gives a method to ensure the constacyclic code satisfying Hermitian dual-containing, and then construct quantum MDS codes (i.e., Maximum Distance Separable codes). This thesis mainly has three parts as follows.Firstly, let p and l be different odd primes,1≠2,3 and e a positive divisor of p-1. We calculate the magnitude of Gauss periods of orders e=3,4,6 over Fl. As their applications, we construct some cyclotomic sequences of orders e=3,4,6, respectively, determine their linear complexity and minimal polynomials, use the minimal polynomials of cyclotomic sequences to construct cyclic codes of length p over Fl. Under certain conditions, we give the lower bounds of minimum distances of the corresponding cyclic codes. When 1=3, we also obtain the linear complexity and minimal polynomials of some classes of generalized cyclotomic sequences.Secondly, let Fq be a finite field with q elements, where q is a power of a prime. Let Fr be an extension field of Fq with r=qm and a a primitive element of Fr. Suppose that n=n1n2 with gcd(n1,n2)=1, n|r-1. Set N1=(r-1)/n1, N2=(r-1)/n2, g1=αN1, g2=αN2, g=g1g2.Define a cyclic code over Fq asc={c(a,b)=(Trr/q(a+b),Trr/q(ag1+bg),…,Trr/q(ag1n-1+bgn-1):a,b∈Fr}, where Trr/q denotes the trace mapping from Fr to Fq. We use the values of exponential sums to give the weight distribution of the cyclic code. In particular, there are a class of two weight cyclic code and a class of three weight cyclic code.Thirdly, quantum MDS codes can be constructed from Hermitian dual-containing constacyclic codes. The thesis gives a method to construct constacyclic codes satisfying Hermitian dual-containing. Using the method, we get not only the well-know parameters of quantum MDS code, but also new parameters of quantum MDS codes.