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Hermitian dual-containing constacyclic BCH codes and related quantum codes of length (q 2m -1)/(q+1).

Xubo Zhao, Xiaoping Li, Qiang Wang, Tongjiang Yan

Published Jul 27, 2020• 0 citations
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Abstract

In this paper, we study a family of constacyclic BCH codes over $\mathbb{F}_{q^2}$ of length $n=\frac{q^{2m}-1}{q+1}$, where $q$ is a prime power, and $m\geq2$ an even integer. The maximum design distance of narrow-sense Hermitian dual-containing constacyclic BCH codes of length $n$ is determined. Furthermore, the exact dimension of the constacyclic BCH codes with given design distance is computed. As a consequence, we are able to derive the parameters of quantum codes as a function of their design parameters of the associated constacyclic BCH codes. This improves the result by Yuan et al. (Des Codes Cryptogr 85(1): 179-190, 2017), showing that with the same lengths, except for three trivial cases ($q=2,3,4$), our resultant quantum codes can always yield strict dimension or minimum distance gains than the ones obtained by Yuan et al.. Moreover, fixing length $n=\frac{q^{2m}-1}{q+1}$, some constructed quantum codes have better parameters or are beneficial complements compared with some known results (Aly et al., IEEE Trans Inf Theory 53(3): 1183-1188, 2007, Li et al., Quantum Inf Process 18(5): 127, 2019, Wang et al., Quantum Inf Process 18(8): 323, 2019, Song et al., Quantum Inf Process 17(10): 1-24, 2018.).

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In this paper, we study a family of constacyclic BCH codes over Fq2\mathbb{F}_{q^2}Fq2​ of length n=q2m−1q+1n=\frac{q^{2m}-1}{q+1}n=q+1q2m−1​, where qqq is a prime power, and m≥2m\geq2m≥2 an even integer. The maximum design distance of narrow-sense Hermitian dual-containing constacyclic BCH codes of length nnn is determined. Furthermore, the exact dimension of the constacyclic BCH codes with given design distance is computed. As a consequence, we are able to derive the parameters of quantum codes as a function of their design parameters of the associated constacyclic BCH codes. This improves the result by Yuan et al. (Des Codes Cryptogr 85(1): 179-190, 2017), showing that with the same lengths, except for three trivial cases (q=2,3,4q=2,3,4q=2,3,4), our resultant quantum codes can always yield strict dimension or minimum distance gains than the ones obtained by Yuan et al.. Moreover, fixing length n=q2m−1q+1n=\frac{q^{2m}-1}{q+1}n=q+1q2m−1​, some constructed quantum codes have better parameters or are beneficial complements compared with some known results (Aly et al., IEEE Trans Inf Theory 53(3): 1183-1188, 2007, Li et al., Quantum Inf Process 18(5): 127, 2019, Wang et al., Quantum Inf Process 18(8): 323, 2019, Song et al., Quantum Inf Process 17(10): 1-24, 2018.).
Hermitian dual-containing constacyclic BCH codes and related quantum codes of length q2m−1q+1\frac{q^{2m}-1}{q+1}q+1q2m−1​
In this paper, we study a family of constacyclic BCH codes over Fq2\mathbb{F}_{q^2}Fq2​ of length n=q2m−1q+1n=\frac{q^{2m}-1}{q+1}n=q+1q2m−1​, where qqq is a prime power, and m≥2m\geq2m≥2 an even integer. The maximum design distance of narrow-sense Hermitian dual-containing constacyclic BCH codes of length nnn is determined. Furthermore, the exact dimension of the constacyclic BCH codes with given design distance is computed. As a consequence, we are able to derive the parameters of quantum codes as a function of their design parameters of the associated constacyclic BCH codes. This improves the result by Yuan et al. (Des Codes Cryptogr 85(1): 179-190, 2017), showing that with the same lengths, except for three trivial cases (q=2,3,4q=2,3,4q=2,3,4), our resultant quantum codes can always yield strict dimension or minimum distance gains than the ones obtained by Yuan et al.. Moreover, fixing length n=q2m−1q+1n=\frac{q^{2m}-1}{q+1}n=q+1q2m−1​, some constructed quantum codes have better parameters or are beneficial complements compared with some known results (Aly et al., IEEE Trans Inf Theory 53(3): 1183-1188, 2007, Li et al., Quantum Inf Process 18(5): 127, 2019, Wang et al., Quantum Inf Process 18(8): 323, 2019, Song et al., Quantum Inf Process 17(10): 1-24, 2018.).
Letqqqbe a prime power andm≥3m\geq 3m≥3be odd. Suppose thatn=q2m−1an=\frac {q^{2m}-1}{a}n=aq2m−1​witha∣(qm+1)a|(q^{m}+1)a∣(qm+1)and3≤a≤2(q2−q+1)3\leq a \leq 2(q^{2}-q+1)3≤a≤2(q2−q+1). This paper mainly determines the actual maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes overFq2\mathbb {F}_{q^{2}}Fq2​of lengthnnn. Firstly, we give the maximum designed distanceδm,aR\delta _{m,a}^{R}δm,aR​of narrow-sense Hermitian dual-containing BCH codes. Secondly, we show that there are also non-narrow-sense ones of designed distance up toδm,aR\delta _{m,a}^{R}δm,aR​. It is worth mentioning that our maximum designed distanceδm,aR>⌈a2⌉δmA\delta _{m,a}^{R}>\lceil \frac {a}{2}\rceil \delta _{m}^{A}δm,aR​>⌈2a​⌉δmA​, whereδmA\delta _{m}^{A}δmA​is given by Alyet al.(IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183-1188, 2007). Thus, many families of Hermitian dual-containing BCH codes with relatively large designed distance are obtained. Using the Hermitian construction to them, we can subsequently construct different classes of nonprimitive quantum codes, which are new in the sense that their parameters are not covered in the literature.
By studying the properties of [Formula: see text]-cyclotomic cosets, the maximum designed distances of Hermitian dual-containing constacyclic Bose–Chaudhuri–Hocquenghem (BCH) codes with length [Formula: see text] are determined, where [Formula: see text] is an odd prime power and [Formula: see text] is an integer. Further, their dimensions are calculated precisely for the given designed distance. Consequently, via Hermitian Construction, many new quantum codes could be obtained from these codes, which are not covered in the literature.
Let [Formula: see text] be an odd prime power and [Formula: see text] be a positive integer. Maximum designed distance such that negacyclic BCH codes over [Formula: see text] of length [Formula: see text] are Hermitian dual-containing codes is given. The dimension of such Hermitian dual-containing negacyclic codes is completely determined by analyzing cyclotomic cosets. Quantum negacyclic BCH codes of length [Formula: see text] are obtained by using Hermitian construction. The constructed quantum negacyclic BCH codes produce new quantum codes with parameters better than those obtained from quantum BCH codes.
Let q ≥ 3 be odd prime power. Maximal designed distances of non-primitive Hermitian dual containing q2- ary BCH codes of length n = (q4-1)/2 and 2(q2-1) are determined. For each given n, a group of non-narrow-sense Hermitian dual containing BCH codes are constructed and their dimensions are computed. Using these non-narrow-sense BCH codes, many new quantum codes are constructed. Our maximal designed distances improve upon the known results in the literature and many of new quantum BCH codes are better than those constructed from narrow-sense BCH codes.