Parameters and characterizations of hulls of some projective narrow-sense BCH codes

Yuwen Huang, Chengju Li, Qi Wang, Zongrun Du

Abstract: 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 $$\mathbb F_q$$ F q be the finite field of order q and $$n = \frac{q^m-1}{q-1}$$ n = q m - 1 q - 1 , where q is a power of a prime and $$m \ge 2$$ m ≥ 2 is an integer. Let $${\mathcal {C}}_{(q,n,\delta )}$$ C ( q , n , δ ) be a projective narrow-sense BCH code over $$\mathbb F_q$$ F q with designed distance $$\delta $$ δ . In this paper, we will investigate both the dimensions and the minimum distances of $$\text {Hull}({\mathcal {C}}_{(q,n,\delta )})$$ Hull ( C ( q , n , δ ) ) , where $$2 \le \delta \le \frac{2(q^{\frac{m+1}{2}} -1)}{q-1}$$ 2 ≤ δ ≤ 2 ( q m + 1 2 - 1 ) q - 1 if $$m \ge 5$$ m ≥ 5 is odd and $$2 \le \delta \le \frac{q^{\frac{m}{2}+1}-1}{q-1}-q+1$$ 2 ≤ δ ≤ q m 2 + 1 - 1 q - 1 - q + 1 if $$m \ge 6$$ m ≥ 6 is even. As a byproduct, a sufficient and necessary condition on the Euclidean dual-containing BCH code $${\mathcal {C}}_{(q,n,\delta )}$$ C ( q , n , δ ) is documented. In addition, we present some characterizations of the hulls of ternary projective narrow-sense BCH codes when $$\dim \Big (\text {Hull} ({\mathcal {C}}_{(3,n,\delta )})\Big )=k-1, \ k-2$$ dim ( Hull ( C ( 3 , n , δ ) ) ) = k - 1 , k - 2 for even $$m \ge 2$$ m ≥ 2 ; and $$\dim \Big (\text {Hull} ({\mathcal {C}}_{(3,n,\delta )})\Big )=k-1, \ k-2m-1$$ dim ( Hull ( C ( 3 , n , δ ) ) ) = k - 1 , k - 2 m - 1 for odd $$m\ge 3$$ m ≥ 3 , where k is the dimension of $${\mathcal {C}}_{(3,n,\delta )}$$ C ( 3 , n , δ ) .