Relative Hulls and Quantum Codes

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.