Let $R=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+\cdots+u^{k}\mathbb{F}_{2^{m}}$, where $\mathbb{F}_{2^{m}}$ is a finite field with $2^{m}$ elements, $m$ is apositive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, wepropose the constructions of two new families of quantum codes obtained fromdual-containing cyclic codes of odd length over $R$. A new Gray map over $R$ isdefined and a sufficient and necessary condition for the existence ofdual-containing cyclic codes over $R$ is given. A new family of $2^{m}$-aryquantum codes is obtained via the Gray map and the Calderbank-Shor-Steaneconstruction from dual-containing cyclic codes over $R.$ Furthermore, a newfamily of binary quantum codes is obtained via the Gray map, the trace map andthe Calderbank-Shor-Steane construction from dual-containing cyclic codes over$R.$
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机译：令$ R = \ mathbb {F} _ {2 ^ {m}} + u \ mathbb {F} _ {2 ^ {m}} + \ cdots + u ^ {k} \ mathbb {F} _ {2 ^ {m}} $，其中$ \ mathbb {F} _ {2 ^ {m}} $是包含$ 2 ^ {m} $个元素的有限字段，$ m $是一个正整数，$ u $是一个不确定的$ u ^ {k + 1} = 0. $在本文中，我们提出了两个新的量子码族的构造，这些量子码是从奇数长度超过$ R $的含双循环码中获得的。定义了$ R $上的新Gray映射，并给出了$ R $上包含偶对的循环码存在的充分必要条件。通过格雷映射和$ R上含双重循环码的Calderbank-Shor-Steane构造，获得了新的$ 2 ^ {m} $元量子码家族。此外，通过格雷获得了新的二元量子码家族地图，跟踪地图和来自$ R. $的双重包含循环代码的Calderbank-Shor-Steane构造
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