Good Entanglement-Assisted Qubit Codes from Matrix Product Codes

We study the Hermitian hulls of matrix product codes and use them to design linear codes with arbitrary hull dimensions. We continue by looking into some propagation rules that preserve the hull dimensions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{F}_{4}$</tex> -linear codes. We propose a recursive method to keep the Hermitian hull dimension fixed while increasing the dimensions or the minimum distances of codes built from a given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{F}_{4}$</tex> -linear code. Using the Hermitian construction route, we derive good entanglement-assisted quantum codes from matrix product codes and recursively iterated codes.