With entanglement-assisted (EA) formalism, arbitrary classical linear codes are allowed to transform into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, based on classical cyclic MDS codes by exploiting pre-shared maximally entangled states, we construct two families of $q$-ary entanglement-assisted quantum MDS codes $[[\frac{q^{2}+1}{a},\frac{q^{2}+1}{a}-2(d-1)+c,d;c]]$, where q is a prime power in the form of $am+l$, and $a=(l^2+1)$ or $a=\frac{(l^2+1)}{5}$. We show that all of $q$-ary EAQMDS have minimum distance upper limit much larger than the known quantum MDS (QMDS) codes of the same length. Most of these $q$-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.

中文翻译：

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循环码中的两个纠缠辅助量子MDS码系列

使用纠缠辅助（EA）形式，通过使用发送者和接收者之间的预共享纠缠，可以将任意经典线性代码转换为EAQECC。在本文中，我们基于经典循环MDS代码，通过利用预共享的最大纠缠态，构造了两个$ q $元纠缠辅助量子MDS代码$ [[\ frac {q ^ {2} +1} { a}，\ frac {q ^ {2} +1} {a} -2（d-1）+ c，d; c]] $，其中q是$ am + l $形式的素数，和$ a =（l ^ 2 + 1）$或$ a = \ frac {（l ^ 2 + 1）} {5} $。我们显示，所有$ q $ ary EAQMDS的最小距离上限远大于相同长度的已知量子MDS（QMDS）码。这些$ q $ ary EAQMDS代码中的大多数都是新的，因为它们的参数未包含在文献中的代码中。