Summary: | Generally, it is not easy to construct quantum maximal-distance-separable (MDS) codes with the minimum distance greater than <inline-formula> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula>. The minimum distance of quantum MDS codes can achieve <inline-formula> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula> or exceed <inline-formula> <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> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula> or exceed <inline-formula> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula> by utilizing the decomposition of the defining set and <inline-formula> <tex-math notation="LaTeX">$q^{2}$ </tex-math></inline-formula>-cyclotomic cosets of constacyclic codes with length <inline-formula> <tex-math notation="LaTeX">$\frac {q^{2}+1}{\gamma }$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\gamma =t^{2}+1$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> is a power of 2 and <inline-formula> <tex-math notation="LaTeX">$q=t^{e}>4$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$e\equiv 1~\textrm {mod}~4$ </tex-math></inline-formula> or <inline-formula> <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> <tex-math notation="LaTeX">$\frac {q}{2}+1$ </tex-math></inline-formula>.
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