Two Kinds Of Hulls Of BCH Codes And Their Parameters

Self-orthogonal codes are an interesting type of linear codes due to their wide appli-cations in communication and cryptography.Especially,for quantum communication and quantum computing,self orthogonal codes play an important role as an important tool to characterize quantum error-correcting codes.In this paper,two kinds of self-orthogonal codes with special dimensions are determined by characterizing BCH codes.As a special cyclic code,BCH code inherits the excellent properties of simple encoding and decoding circuit and strong error correction ability.In addition,there is a close relationship be-tween the minimum distance of BCH codes and their generating polynomials.The codes can be constructed according to the requirements of error correction ability.Moreover,the coding and decoding algorithms are very efficient and widely used in communication and storage systems.The hull of BCH code is defined as the intersection of BCH code C and its dual code C~⊥.It is not difficult to prove that the hull is self-orthogonal.In this paper,we investigate two classes of the hull of BCH codes by characterizing BCH codes,and then present several classes of self-orthogonal codes.The main work of this paper are given as follows.·The general characterizations of cyclic codes having dimHull(C)=k-1 or k~⊥-1 were presented by employing their defining sets.·The sufficient and necessary conditions for some primitive binary and ternary BCHcodes to have(k-1)-dimensional hulls were developed by giving the lower andupper bounds on their designed distances.·Several classes of self-orthogonal codes were obtained from the hulls of these primi-tive codes and their parameters were investigated.The dimensions and lower bounds on minimum distances of these self-orthogonal codes were presented,and the ex-plicit minimum distances were given in the binary case.·The sufficient and necessary conditions for some primitive and projective BCH codes to have(k~⊥-1)-dimensional hulls also were documented and demonstrated in detail.In particular,we discuss different finite fields more extensively in this part,not only in binary and ternary fields.Furthermore,we give some examples of the codes C~⊥and Hull(C),most of which are optimal.