Parameters And Characterizations Of Hulls Of Some Projective Narrow-sense BCH Codes

As a special class of linear codes,BCH codes can correct multiple random errors and have an efficient decoding algorithm.The hull of a linear code is defined to be the intersection of the code and its dual.It is obvious that the hulls are self-orthogonal codes.It is known that self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography.Self-orthogonal codes can also construct quantum error correcting codes(QECC)and entanglement-assisted quantum error correcting codes(EAQECC).Thus the investigation of hull is important and interesting.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.Based on the good algebraic structure and properties of BCH codes,we will investigate the hulls of BCH codes in this paper.Let Fq be the finite field of order q and n=qm-1/q-1,where a is a power of a prime and m>2 is an integer.Let C(q,n,?)be a projective narrow-sense BCH code over Fq with designed distance ?,and Hull(C(q,n,?)be the hull of C(q,n,?).In this paper,we will investigate the dimensions and give the lower bounds on the minimum distances of the Hull(C(q,n,?)),and characterize the BCH codes of which hulls have fixed dimensions.The main works of this paper are as follows:1.By investigating the defining set and cyclotomic cosets,we get both the dimensions and the minimum distances of Hull(C(q,n,?),where 2 ???2(qm+1/2-1)/q-1 if m?5 is odd and 2 ???qm/2+1-1/q-1-q+1 if m?6 is even.As a byproduct,a sufficient and necessary condition on the Euclidean dual-containing BCH code C(q,n,?)is documented.2.We present some characterizations of the hulls of ternary projective narrow-sense BCH codes when dim(Hull(C(3,n,?)))=k-1,k-2,k-2m-1 for m ? 2 by investigating the defining set.The necessary and sufficient conditions of the upper and lower bounds of the design distance are given,and we give some examples of the optimal codes.3.We construct the entanglement-assisted quantum error correcting codes by the hulls of projective narrow-sense BCH codes C(q,n,?),and some examples are given.