  Given two $q$-ary codes $C_1$ and $C_2$, the relative hull of $C_1$ with
respect to $C_2$ is the intersection $C_1\cap C_2^\perp$. We prove that when
$q>2$, the relative hull dimension can be repeatedly reduced by one, down to a
certain bound, by replacing either of the two codes with an equivalent one. The
reduction of the relative hull dimension applies to hulls taken with respect to
the $e$-Galois inner product, which has as special cases both the Euclidean and
Hermitian inner products. We give conditions under which the relative hull
dimension can be increased by one via equivalent codes when $q>2$. We study
some consequences of the relative hull properties on entanglement-assisted
quantum error-correcting codes and prove the existence of new
entanglement-assisted quantum error-correcting maximum distance separable
codes, meaning those whose parameters satisfy the quantum Singleton bound.
