On Hulls Of Some Primitive BCH Codes And Their Parameters

Error-correcting codes ensure the reliability of communication system due to their ability of error detection and error correction.Self-orthogonal codes play an important role in error-correcting codes and they are widely used in communication,data storage and cryptography.With the development of quantum communication technology,many researchers pay much attention to self-orthogonal codes since they can be used to construct quantum error-correcting codes.It is known that self-orthogonal codes can be constructed from hulls of linear codes.As a special class of linear codes,BCH codes have good structure and strong ability of error correction.In addition,they can be constructed flexibly.Therefore,we deeply investigate the hulls of BCH codes and construct several classes of self-orthogonal codes with large dimensions.The hull is defined to be the intersection of a linear code C and its dual code.In this thesis,we consider the hulls defined by Euclidean and Hermitian inner product,respectively.We present the dimensions of the hulls of BCH codes by employing defining set and the lower bounds on their minimum distances by employing the BCH bound.Furthermore,we determine the minimum distances in some special cases.The main contributions of this thesis are given as follows:·By restricting the designed distance ? of BCH code C(Q,Qm-1,?,b),we give some sufficient and necessary conditions that the Hermitian hull of C(Q,Qm-1,?,b)has dimensions k-1,k-2,and k?H-1,and a sufficient condition that the Hermitian hull has dimension k-3.We give the lower bounds on their minimum distances,and determine their explicit values in some special cases.·When 2??<Q2-1,we present the dimension of the Hermitian hull of BCH code C(Q,Q2-1,?,1).Furthermore,we give a sufficient and necessary condition by using the disigned distance ? such that the dimension of the Hermitian hull is maximal,and obtain the lower bound on the minimum distance of the hull by employing the BCH bound.·When 2???qm-1 and m=2,3,we present the dimensions of the Euclidean hulls of BCH codes C(q,qm-1,?,1).We also give two sufficient and necessary conditions by using the disigned distance ? such that the dimensions of the Euclidean hulls are maximal,and obtain the lower bounds on the minimum distances of the hulls by employing the BCH bound.·We determine the dimensions and the lower bounds on the minimum distances of BCH codes C(Q,Q2-1,?,1),C(q,q2-1,?,1)and C(q,q3-1,?,1).