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Modern Physics Letters B

New optimal asymmetric quantum codes from constacyclic codes

Guanghui Zhang, Bocong Chen, Liangchen Li

Published Jun 17, 2014• 14 citations
Publisher

Abstract

In this paper, we construct two classes of asymmetric quantum codes by using constacyclic codes. The first class is the asymmetric quantum codes with parameters [[q2 + 1, q2 + 1 - 2(t + k + 1), (2k + 2)/(2t + 2)]]q2 where q is an odd prime power, t, k are integers with [Formula: see text], which is a generalization of [J. Chen, J. Li and J. Lin, Int. J. Theor. Phys. 53 (2014) 72, Theorem 2] in the sense that we do not assume that q ≡1 ( mod 4). The second one is the asymmetric quantum codes with parameters [Formula: see text], where q ≥ 5 is an odd prime power, t, k are integers with 0 ≤ t ≤ k ≤ q - 1. The constructed asymmetric quantum codes are optimal and their parameters are not covered by the codes available in the literature.

DOI
10.1142/s0217984914501267

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In this paper, we construct two classes of asymmetric quantum codes by using constacyclic codes. The first class is the asymmetric quantum codes with parameters [[q2 + 1, q2 + 1 - 2(t + k + 1), (2k + 2)/(2t + 2)]]q2 where q is an odd prime power, t, k are integers with [Formula: see text], which is a generalization of [J. Chen, J. Li and J. Lin, Int. J. Theor. Phys. 53 (2014) 72, Theorem 2] in the sense that we do not assume that q ≡1 ( mod 4). The second one is the asymmetric quantum codes with parameters [Formula: see text], where q ≥ 5 is an odd prime power, t, k are integers with 0 ≤ t ≤ k ≤ q - 1. The constructed asymmetric quantum codes are optimal and their parameters are not covered by the codes available in the literature.
In this paper, we propose the construction of asymmetric quantum codes from two families of constacyclic codes over finite field [Formula: see text] of code length [Formula: see text], where for the first family, [Formula: see text] is an odd prime power with the form [Formula: see text] ([Formula: see text] is integer) or [Formula: see text] ([Formula: see text] is integer) and [Formula: see text]; for the second family, [Formula: see text] is an odd prime power with the form [Formula: see text] or [Formula: see text] ([Formula: see text] is integer) and [Formula: see text]. As a result, families of new asymmetric quantum codes [Formula: see text] with [Formula: see text] distance larger than [Formula: see text] are obtained, which are not covered by the asymmetric quantum error-correcting codes (AQECCs) in Refs. 32 and 33 [J.-Z. Chen, J.-P. Li and J. Lin, Int. J. Theor. Phys. 53, 72 (2014); L. Wang and S. Zhu, Int. J. Quantum Inf. 12, 1450017 (2014)] that [Formula: see text]. Also, all the newly obtained asymmetric quantum codes are optimal according to the singleton bound for asymmetric quantum codes.
In this paper, we construct two classes of new quantum maximum-distance-separable (MDS) codes with parameters [Formula: see text], where q is an odd prime power with q ≡ 3 (mod 4) and [Formula: see text]; [[8(q - 1), 8(q - 1) - 2d + 2, d]]q, where q is an odd prime power with the form q = 8t - 1 (t is an even positive integer) and [Formula: see text]. Comparing the parameters with all known quantum MDS codes, the quantum MDS codes exhibited here have minimum distances bigger than the ones available in the literature.
Assume that qqq is a prime power and m≥2m\geq 2m≥2 is a positive integer. Cyclic codes over Fq2m\mathbb{F}_{q^{2m}}Fq2m​ of length n=q2m−1ρn=\frac{q^{2m}-1}{\rho }n=ρq2m−1​ with ρ∣(q−1)\rho\mid (q-1)ρ∣(q−1), and constacyclic codes over Fq2m\mathbb{F}_{q^{2m}}Fq2m​ of length n=q2m−1ρn=\frac{q^{2m}-1}{\rho }n=ρq2m−1​ with ρ∣(q+1)\rho\mid (q+1)ρ∣(q+1) are considered in this paper, respectively. Two classes of quantum codes are derived from the images of these codes by the Hermitian construction. Compared with the previously known quantum codes, the quantum codes in our scheme have better parameters.