The Distances And Weight Distributions Of Several Classes Of Codes

With the rapid development of internet technology,more and more attention is paid to the validity,the confidentiality and the correctness of information transmis-sion.Channel coding is a theory developed to solve information errors in transmis-sion.Due to the diversity of errors in information transmission,various distances have been introduced into coding theory,such as Hamming distance,symbol-pair distance,b-distance and so on.The weight distributions of linear codes determine their error-corre.cting ability and their decoding efficiency.Therefore,the research on the distances and the weight distributions of codes have very important theoret-ical significance and practical implementations.In this dissertation,the distances and the weight distributions of several classes of constacyclic codes,cyclic codes and linear codes are studied.The main results are given as follows.In Chapter 3,let 1?b?[p/2],the b-distances of constacyclic codes of length ps and MDS b-symbol constacyclic codes of this length are determined,where s is a positive integer and p is a prime.Our results generalize the results about Hamming distances and symbol-pair distances of constacyclic codes of length ps in References[39],[49]and[1071.In Chapter 4,we determine the symbol-pair distances of constacyclic codes of length 2ps and MDS symbol-pair constacyclic codes of this length,where s is a positive integer,p is an odd prime.In addition,some MDS symbol-pair cyclic codes of special lengths with symbol-pair distances of 6 or 7 are const.ructed.In Chapter 5.we first show the relationships of Hamming distances between the single-root constacyclic codes and the repeated-root constacyclic codes,and then using this result,the degrees of generator polynomials of all optimal repeated-root constacyclic codes with respect to the Singleton bound are given.Furthermore,we obtain the Hamming distances of constacyclic codes of length 3ps and the optimal cases of these codes with respect to the Singleton bound and the Griesmer bound in terms of the Hamming distances,where s is an integer,p?3 is a prime.In Chapter 6,let p be an odd prime,k be a positive integer,? be a primitive element of finite field Fp2m.Using the connection between the type and the rank of quadratic form,we determine the weight distribution of cyclic code In addition,assume that f(x)is a special function over finite field Fpm,let be a defining set.We obtain the weight distribution of linear code and some optimal codes with respect to the Singleton bound and the Griesmer bound.In Chapter 7,we discuss the algebraic structures of(?+u?)-constacyclic codes of arbitrary length over Fpm[u]/<ul>and their duals,where l is a positive integer such that ul=0,p is a prime,??Fpm*and ??Fpm*+uFpm+…+ul-2Fpm.Based on the algebraic structures,when the generator polynomials of these codes are binomial polynomials,the b-distances and the homogeneous distances of(a+u?)-constacyclic codes are determined.