Parameters and characterizations of hulls of some projective narrow-sense BCH 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 cryptography. Let F_q be the finite field of order q and n=qm-1q-1, where q is a power of a prime and m≥ 2 is an integer. Let C_(q,n,δ) be a projective narrow-sense BCH code over F_q with designed distance δ. In this paper, we will investigate both the dimensions and the minimum distances of Hull (C_(q,n,δ)) , where 2≤δ≤2(qm+12-1)q-1 if m≥ 5 is odd and 2≤δ≤qm2+1-1q-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,δ) is documented. In addition, we present some characterizations of the hulls of ternary projective narrow-sense BCH codes when dim(Hull(C(3,n,δ)))=k-1,k-2 for even m≥ 2 ; and dim(Hull(C(3,n,δ)))=k-1,k-2m-1 for odd m≥ 3 , where k is the dimension of C_(3,n,δ).