# New quantum codes from dual-containing cyclic codes over finite rings

New quantum codes from dual-containing cyclic codes over finite rings Let $$R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}$$ R = F 2 m + u F 2 m + ⋯ + u k F 2 m , where $$\mathbb {F}_{2^{m}}$$ F 2 m is the finite field with $$2^{m}$$ 2 m elements, m is a positive integer, and u is an indeterminate with $$u^{k+1}=0.$$ 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 $$2^{m}$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# New quantum codes from dual-containing cyclic codes over finite rings

, Volume 15 (11) – Aug 22, 2016
12 pages

/lp/springer_journal/new-quantum-codes-from-dual-containing-cyclic-codes-over-finite-rings-kYjgaW70r0
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-016-1426-5
Publisher site
See Article on Publisher Site

### Abstract

Let $$R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}$$ R = F 2 m + u F 2 m + ⋯ + u k F 2 m , where $$\mathbb {F}_{2^{m}}$$ F 2 m is the finite field with $$2^{m}$$ 2 m elements, m is a positive integer, and u is an indeterminate with $$u^{k+1}=0.$$ 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 $$2^{m}$$ 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.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Aug 22, 2016

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