Research On Constacyclic BCH Codes And Their Applications

With the continuous development of information technology,error-correcting codes play an increasingly important role in information security.Selecting specific codes according to actual needs is a key problem in the theory of error-correcting codes,and the most used cyclic codes are a hot topic.Cyclic codes not only have efficient circuit implementation,but also have been widely used in mobile communications,radar,aerospace and other fields.Constacyclic codes are the generalization and development of cyclic codes.They not only inherit the good characteristics of cyclic codes,but also have more flexible parameters.However,compare with the study of cyclic codes,constacyclic codes need further research and development.Especially constacyclic BCH codes,which have good error correcting capability and can be easily encoded.Therefore,it is necessary to study constacyclic codes thoroughly and systematically.This dissertation investigates the parameters of constacyclic codes,in particular constacyclic BCH codes.Based on constacyclic codes,quantum codes with good parameters,MDS symbol-pair codes and optimal locally repairable codes are constructed,respectively.The specific research contents of this dissertation are described as follows:1)The minimum distance of repeated-root constacyclic codes is studied.By analyzing the relationship between constacyclic codes and cyclic codes,a relationship between the minimum Hamming distance of repeated-root constacyclic codes and that of simple-root constacyclic codes is given.By the properties of polynomial,the minimum pair-distances of two classes of repeated-root constacyclic codes are completely determined,and a class of MDS symbol-pair codes is obtained.2)The parameters of constacyclic BCH codes are studied.Through the theory of cyclotomic cosets,the following results are obtained:Firstly,the dimension of a class of BCH codes is determined.Secondly,the first few largest q-cyclotomic coset leaders modulo(qm-1)/2 are given.Via the quadratic form theory over finite fields,the weight distributions of four classes of BCH codes are completely determined.The dimension of a class of negacyclic BCH codes is also determined,and the obtained results are applied to the construction of quantum codes.A class of quantum codes with good parameters is obtained.Finally,the weight distributions of a class of q-ary BCH codes with length(qm-1)/(q-1)and a class of q-ary negacyclic BCH codes with length(q2m-1)/(q-1)are studied,respectively.3)The constructions of locally repairable codes are studied.Firstly,an explicit construction of optimal locally repairable codes with locality r(r locally repairable codes)is given via repeated-root cyclic codes,and several classes of optimal r locally repairable codes with unbounded length and minimum distance 4 are constructed.Secondly,an explicit construction of optimal locally repairable codes with locality(r,?)((r,?)locally repairable codes)is given via constacyclic codes,and several classes of optimal(r,?)locally repairable codes with unbounded length and minimum distance<2? are constructed.