New quantum codes from dual-containing cyclic codes over finite rings |
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Authors: | Yongsheng Tang Shixin Zhu Xiaoshan Kai Jian Ding |
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Affiliation: | 1.School of Mathematics and Statistics,Hefei Normal University,Hefei,China;2.School of Mathematics,Hefei University of Technology,Hefei,China;3.Department of Common Course,Anhui Xinhua University,Hefei,China |
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Abstract: | Let \(R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}\), where \(\mathbb {F}_{2^{m}}\) is the finite field with \(2^{m}\) elements, m is a positive integer, and u is an indeterminate with \(u^{k+1}=0.\) In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined, and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of \(2^{m}\)-ary quantum codes is obtained via the Gray map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R. |
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