Relative Hulls and Quantum Codes

Given two <tex-math notation="LaTeX">q </tex-math> -ary codes <tex-math notation="LaTeX">C₁ </tex-math> and <tex-math notation="LaTeX">C₂ </tex-math> , the relative hull of <tex-math notation="LaTeX">C₁ </tex-math> with respect to <tex-math notation="LaTeX">C₂ </tex-math> is the intersection <tex-math notation="LaTeX">C₁∩ C₂⊥ </tex-math> . We prove that when <tex-math notation="LaTeX">$q>2$ </tex-math> , 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 <tex-math notation="LaTeX">e </tex-math> -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 <tex-math notation="LaTeX">$q>2$ </tex-math> . 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.