The Dual Codes Of Primitive And Projective Narrow-Sense BCH Codes

As a special subclass of cyclic codes,BCH codes have wide applications in communication and storage systems.BCH codes have strong ability of error correction.There are few results on the dual codes of BCH codes.This thesis aims to characterize the parameters of the dual codes of BCH codes by studying the definition set of BCH code and its dual code.Let F_q be the finite field of order q,where q is a power of a prime and n is an integer.A BCH code of length n over F_q is always relative to an n-th primitive root of unity β in an extension field of F_q,and is called a dually-BCH code if its dual is also a BCH code relative to the same β.The question as to whether a BCH code is a dually-BCH code is in general very hard to answer.In this thesis,the main research content is dually-BCH code.The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent.In addition,the hull of a linear code is defined to be the intersection of the code and its dual code.It is obvious that the hulls are self-orthogonal codes.Based on dually-BCH codes,this thesis studies parameters of hulls of primitive binary narrowsence BCH codes and their subcodes.The main contributions of this thesis are given as follows:1.Sufficient and necessary conditions for C_{（q,q^m-1,δ）}and C_{（3,3m-1/2,δ）}being dually-BCH were developed.The parameters of the narrow-sense BCH codes and their duals were investigated.Some lower bounds on the minimum distances of the dual codes of primitive and projective narrow-sense BCH codes were developed.Especially for binary primitive narrow-sense BCH codes,the bounds on minimum distances of the dual codes improve the classical Sidel’nikov bound,and are also better than the Carlitz-Uchiyama bound for large designed distances δ,which are classic conclusions more than 50 years ago.2.The dimension of Hull(C_（2,n,δ）)was presented when 2 ≤δ ≤2^[m/2]+1+2^[m/2]-1-1 with m ≥ 5.A new lower bound on the minimum distance of Hull(C_（2,n,δ）)was documented for 2 ≤δ<n.As a byproduct,the dimension of C（2,n,δ）was given when 2 m/2+1 ≤δ≤2^m/2+1+2^m/2-1-1 for even m≥ 6.As a subcode of Hull(C_（2,n,δ）),the parameters of the self-orthogonal code C’ were investigated.