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International Journal of Quantum Information

Quantum codes from cyclic codes over 𝔽q+v𝔽q+v2𝔽q+v3𝔽q

Jian Gao

Published Dec 1, 2015• 41 citations
Publisher

Abstract

We give a construction of quantum codes over [Formula: see text] from cyclic codes over a finite non-chain ring [Formula: see text], where [Formula: see text], p is a prime, [Formula: see text] and [Formula: see text].
DOI
10.1142/s021974991550063x

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We give a construction of quantum codes over Fq{\mathbb{F}_q}Fq​ from cyclic codes over a finite non-chain ring Fq+vFq+v2Fq+v3Fq+v4Fq{\mathbb{F}_q} + v{\mathbb{F}_q} + {v^2}{\mathbb{F}_q} + {v^3}{\mathbb{F}_q} + {v^4}{\mathbb{F}_q}Fq​+vFq​+v2Fq​+v3Fq​+v4Fq​, where q=pr, p is an odd prime, 4|p−1 and v5=v.
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.
Let [Formula: see text] be a prime power and let [Formula: see text] be a finite non-chain ring, where [Formula: see text], are polynomials, not all linear, which split into distinct linear factors over [Formula: see text]. We characterize constacyclic codes over the ring [Formula: see text] and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials [Formula: see text], [Formula: see text], is irrelevant while constructing quantum codes from constacyclic codes over [Formula: see text], it depends only on their degrees.