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 = q(m)- 1/q-1, where q is a power of a prime andm >= 2 is an integer. Let C-(q,C-n,C-delta) be a projective narrow-sense BCH code over F-q with designed distance delta. In this paper, we will investigate both the dimensions and the minimum distances of Hull(C-(q,C-n,C-delta)), where 2 <= delta <= 2(q(m+1/2) -1)/q-1 if m >= 5 is odd and 2 <= delta <= q(m/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, d) is documented. In addition, we present some characterizations of the hulls of ternary projective narrow-sense BCH codes when dim equivalent to Hull(C-(3,C-n,C-delta)) = k - 1, k - 2 for even m >= 2; and dim (Hull(C-(3,C-n,C-delta)) equivalent to k - 1, k - 2m - 1 for odd m >= 3, where k is the dimension of C-(3,C-n,C-delta).