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.
The Bose–Chaudhuri–Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of [Formula: see text]-ary BCH codes with length [Formula: see text] (also called antiprimitive BCH codes in the literature), where [Formula: see text] is a power of 2 and [Formula: see text]. By a detailed analysis of some useful properties about [Formula: see text]-ary cyclotomic cosets modulo [Formula: see text], Hermitian dual-containing conditions for a family of non-narrow-sense antiprimitive BCH codes are presented, which are similar to those of [Formula: see text]-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes.
In this paper, the properties of q2-ary cyclotomic cosets modulo 2n are investigated in detail, where n=[(q2m-1)/(m(q-1))],q ≡ ±1mod m or q2≡ -1mod m when m is even and q is an odd power prime when m is power of 2. Then, non-narrow-sense negacyclic Bose-Chaudhuri-Hocquenghem (BCH) codes of length n that contain their Hermitian duals are determined. By the Hermitian construction, series of quantum codes are obtained from such negacyclic BCH codes. Compared with quantum codes in the [6], [10], [12], our constructed ones have better parameters.
In this paper, we study a family of constacyclic BCH codes over
Fq2 of length
n=q+1q2m−1, where
q is a prime power, and
m≥2 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=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.).
Let
qbe a prime power and
m≥3be odd. Suppose that
n=aq2m−1with
a∣(qm+1)and
3≤a≤2(q2−q+1). This paper mainly determines the actual maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes over
Fq2of length
n. Firstly, we give the maximum designed distance
δm,aRof narrow-sense Hermitian dual-containing BCH codes. Secondly, we show that there are also non-narrow-sense ones of designed distance up to
δm,aR. It is worth mentioning that our maximum designed distance
δm,aR>⌈2a⌉δmA, where
δmAis 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.
In this paper, we study a family of constacyclic BCH codes over
Fq2 of length
n=q+1q2m−1, where
q is a prime power, and
m≥2 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=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.).