# 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∩ C_2^⊥. 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^' of C_1 (or C_2^' of C_2) such that the dimension of C_1^'∩ C_2^⊥ (or C_1 ∩ C_2^'⊥) is one less than the dimension of C_1∩ C_2^⊥. 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.

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