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International Journal of Quantum Information

QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER $\mathbb{F}_4 + u\mathbb{F}_4$

Xiaoshan Kai, Shixin Zhu

Published Mar 1, 2011• 81 citations
Publisher

Abstract

We give a construction for quantum codes from linear and cyclic codes over [Formula: see text]. We derive Hermitian self-orthogonal codes over [Formula: see text] as Gray images of linear and cyclic codes over [Formula: see text]. In particular, we use two binary codes associated with a cyclic code over [Formula: see text] of odd length to determine the parameters of the corresponding quantum code.

DOI
10.1142/s0219749911007757

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QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER F4+uF4\mathbb{F}_4 + u\mathbb{F}_4F4​+uF4​
We give a construction for quantum codes from linear and cyclic codes over [Formula: see text]. We derive Hermitian self-orthogonal codes over [Formula: see text] as Gray images of linear and cyclic codes over [Formula: see text]. In particular, we use two binary codes associated with a cyclic code over [Formula: see text] of odd length to determine the parameters of the corresponding quantum code.
In the paper, we consider a finite non-chain commutative ring [Formula: see text], where [Formula: see text]. We mainly study the construction of quantum codes from cyclic codes over [Formula: see text]. For this, we obtained self-orthogonal codes over [Formula: see text] as gray images of linear and cyclic codes over [Formula: see text], then these codes over [Formula: see text] correspond to a cyclic code over [Formula: see text] of odd length [Formula: see text] used to determine the parameters of the quantum codes.
We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of 000-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions.
On Quantum Codes Obtained From Cyclic Codes Over F2+uF2+u2F2\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2F2​+uF2​+u2F2​
Let R=F2+uF2+u2F2R=\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2R=F2​+uF2​+u2F2​ be a non-chain finite commutative ring, where u3=uu^3=uu3=u. In this paper, we mainly study the construction of quantum codes from cyclic codes over RRR. We obtained self-orthogonal codes over F2\mathbb{F}_2F2​ as gray images of linear and cyclic codes over RRR. The parameters of quantum codes which are obtained from cyclic code over RRR are discussed.