Characterizations Of Two Kinds Of Hulls Of Binary And Ternary Primitive BCH Codes And Constructions Of Correlated Self-Orthogonal Codes

The（Euclidean）hull of a linear code is defined to be the intersection of the code and its Euclidean dual.It is clear that the hulls are self-orthogonal codes,which are an important type of linear codes due to their wide applications in communication and cryp-tography.It is noted that self-orthogonal codes can be applied to construct quantum er-ror correction codes（QECC）and entanglement-assisted quantum error correction codes（EAQECC）.It is then important and interesting to investigate the hulls of linear codes and obtain the self-orthogonal codes.As a special class of cyclic codes,BCH codes have explicit defining sets that include consecutive integers.This indeed gives an advantage to discuss the relations of cyclotomic cosets,and then the BCH codes are viewed as a good candidate to obtain self-orthogonal codes from their hulls.Let F_qbe the finite field of order q and n=q^m-1,where q is a power of a prime p and m≥2 is an integer.Let C_{（q,n,δ,b）}be a primitive BCH code over F_q,and Hull(C_{（q,n,δ,b）})be the hull of C_{（q,n,δ,b）}.The main contributions of this paper are given as follows:1.A general characterization of cyclic codes having dimHull（C）=k-lwas presented by employing their defining sets.2.The sufficient and necessary conditions for some primitive binary and ternary BCH codes to have（k-2）-dimensional hulls were developed by giving the lower and upper bounds on their designed distances.Several classes of self-orthogonal codes were obtained from their hulls and their parameters were investigated.The dimensions and lower bounds on minimum distances of these codes were presented in some cases.3.The sufficient and necessary conditions for some primitive binary and ternary BCH codes to have（k-3）-dimensional hulls were also proposed by giving the lower and upper bounds on their designed distances.Several classes of self-orthogonal codes were obtained from their hulls and their parameters were investigated.The dimensions and lower bounds on minimum distances of these codes were presented in some cases.