Applications of Constacyclic Codes to Some New Entanglement-Assisted Quantum MDS Codes

Generally, it is not easy to construct quantum maximal-distance-separable (MDS) codes with the minimum distance greater than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula> . The minimum distance of quantum MDS codes can achieve <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula> or exceed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula> by adopting pre-shared entanglement. In this work, some new families of entanglement-assisted quantum MDS codes that satisfy the quantum Singleton bound are constructed and the number of maximally entangled states required is determined to make the minimum distance of some constructed codes achieve <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula> or exceed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula> by utilizing the decomposition of the defining set and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q^{2}$ </tex-math></inline-formula> -cyclotomic cosets of constacyclic codes with length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {q^{2}+1}{\gamma }$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma =t^{2}+1$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> is a power of 2 and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q=t^{e}&gt;4$ </tex-math></inline-formula> with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$e\equiv 1~\textrm {mod}~4$ </tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$e\equiv 3~\textrm {mod}~4$ </tex-math></inline-formula> . Moreover, the parameters of these codes constructed in this paper are more general relative to the ones in the literature and the minimum distance of some codes constructed in this paper is larger than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula> .