Relative hulls and quantum codes

The relative hull of a code $C_1$ with respect to another code $C_2$ is the intersection $C_1\cap C_2^\perp$. We prove that the dimension of the relative hull can always be repeatedly reduced by one by replacing any of the two codes with an equivalent one, down to a specified lower bound. We show how to construct an equivalent code $C_1^\prime$ of $C_1$ (or $C_2^\prime$ of $C_2$) such that the dimension of $C_1^\prime \cap C_2^{\perp}$ (or $C_1 \cap C_2^{\prime\perp}$) is one less than the dimension of $C_1\cap C_2^\perp$. Given codes $C_1$ and $C_2$, we provide a method to specify a code equivalent to $C_2$ which gives a relative hull of any specified dimension, between the difference in dimensions of $C_1$ and $C_2$ and the dimension of the relative hull of $C_1$ with respect to $C_2$. These results apply to hulls taken with respect to the $e$-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We also give conditions under which the dimension of the relative hull can be increased by one via equivalent codes. We study the consequences of the relative hull properties on quantum codes constructed via CSS construction. Finally, we use families of decreasing monomial-Cartesian codes to generate pure or impure quantum codes.