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New Quantum MDS codes from Hermitian self-orthogonal generalised Reed-Solomon codes
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Abstract
It is important task to construct quantum maximum-distance-separable (MDS for short) codes with good parameters. In this paper, by using Hermitian self-orthogonal generalized Reed-Solomon (GRS for short) codes, we construct four new classes of quantum MDS codes. Our quantum MDS codes have flexible parameters. And the minimum distances of our quantum MDS codes can be larger than q/2+1. Furthermore, it turns out that our constructions generalize and improve some previous results.
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The construction of quantum maximum distance separable (MDS for short) codes is one of the hot issues in quantum information theory. As far as we know, researchers have done a lot of constructive work in the construction of quantum MDS codes. However, the known results do not cover all parameters. In this paper, we propose an efficient construction implemented by concatenating two existing quantum MDS codes. Compared to a previous work (Fang and Luo in Quantum Inf Process 19(1):16, 2020), we relax the restrictions of the construction and propose some new quantum MDS codes.
We construct quantum MDS codes with parameters for all , . These codes are shown to exist by proving that there are classical generalised Reed–Solomon codes which contain their Hermitian dual. These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if then there is no generalised Reed–Solomon code which contains its Hermitian dual. We also construct an quantum MDS code, an quantum MDS code and a quantum MDS code, which are the first quantum MDS codes discovered for which , apart from the quantum MDS code derived from Glynn’s code.
It has become common knowledge that constructing q-ary quantum MDS codes with minimum distance bigger than q/2+1 is significantly more difficult than constructing those with minimum distance less than or equal to q/2+1. Despite of various constructions of q-ary quantum MDS codes, all known q-ary quantum MDS codes have minimum distance bounded by q/2+1 except for some lengths. The purpose of the current paper is to provide some new q-ary quantum MDS codes with minimum distance bigger than q/2+1. In this paper, we provide several classes of quantum MDS codes with minimum distance bigger than q/2+1. For instance, some examples in these classes include q-ary -quantum MDS codes for cases: (i) and ; (ii) and ; (iii) and ; and (iv) and .
Quantum MDS codes are an important family of quantum codes. In this paper, using generalized Reed–Solomon codes and Hermitian construction, we construct seven classes of quantum MDS codes. All of them provide large minimum distance and most of them are new in the sense that the parameters of quantum codes are different from all the previously known ones.
Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to construct the quantum MDS codes. We can not only provide a unified explanation of many previous constructions, but also produce some new quantum MDS codes.
Quantum maximal-distance-separable (MDS) codes form an important class of
quantum codes. To get q-ary quantum MDS codes, it suffices to find linear MDS
codes C over satisfying by the
Hermitian construction and the quantum Singleton bound. If
, we say that C is a dual-containing code. Many new
quantum MDS codes with relatively large minimum distance have been produced by
constructing dual-containing constacyclic MDS codes (see \cite{Guardia11},
\cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study on
the existence condition for nontrivial dual-containing constacyclic codes. This
would help us to avoid unnecessary attempts and provide effective ideas in
order to construct dual-containing codes. Several classes of dual-containing
MDS constacyclic codes are constructed and their parameters are computed.
Consequently, new quantum MDS codes are derived from these parameters. The
quantum MDS codes exhibited here have parameters better than the ones available
in the literature.
It has been a great challenge to construct new quantum MDS codes. In
particular, it is very hard to construct quantum MDS codes with relatively
large minimum distance. So far, except for some sparse lengths, all known
q-ary quantum MDS codes have minimum distance less than or equal to q/2+1.
In the present paper, we provide a construction of quantum MDS codes with
minimum distance bigger than q/2+1. In particular, we show existence of
q-ary quantum MDS codes with length and minimum distance d for
any (this result extends those given in \cite{Gu11,Jin1,KZ12}); and
with length and minimum distance d for any if
. Our method is through Hermitian self-orthogonal codes. The main idea
of constructing Hermitian self-orthogonal codes is based on the solvability in
\F_q of a system of homogenous equations over \F_{q^2}.
Classical Bose-Chaudhuri-Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length n can contain its dual code only if its designed distance delta=O(radicn), and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and - consequently - the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters
We define and show how to construct nonbinary quantum stabilizer
codes. Our approach is based on nonbinary error bases. It generalizes
the relationship between self-orthogonal codes over F4 and
binary quantum codes to one between self-orthogonal codes over F(q<sup>2
</sup>) and q-ary quantum codes for any prime power q
The problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field GF (4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits. 1. Introduction The relationship between quantum information and classical information is a subject currently receiving much study. While there are many similarities, there are also substantial differences between the two. Classical information cannot travel faster than light, while quantum information appears to in some circumstances. Classical information can be duplicated, while quantum information cannot [18], [37]. It is well known that classical information can be protected from degradation by the use of classical error-correcting codes [30]. Classical error-correcting codes appear to protect classical information by duplicating it, so because o...
The construction of quantum maximum distance separable (abbreviated to MDS) error-correcting codes has become one of the major concerns in quantum coding theory. In this paper, we further generalize the approach developed in the previous paper, and construct several new classes of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. By employing these classical MDS codes, we obtain several classes of quantum MDS codes with large minimum distance. It turns out that our quantum MDS codes exhibited here have less constraints on the selection of code length and some of them have not been constructed before, and in some cases, have larger minimum distance than previous literature. Meanwhile, about half of the distance parameters of our codes are greater than q2+1.
It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of
q
-ary quantum MDS codes by using generalized Reed–Solomon (GRS) codes and Hermitian construction. The minimum distances of our quantum MDS codes can be larger than
. Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in
[1]
and
[2]
, hence some of their results can be easily derived from ours via the propagation rule. Moreover, some known quantum MDS codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum MDS codes are also improved as well.
Let p be a prime and let q be a power of p. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance- separable (MDS) codes with parameters for any , and for any with . Our quantum codes have flexible parameters, and have minimum distances larger than when . Furthermore, it turns out that our constructions generalize and improve some previous results.
One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature.
Quantum maximum-distance-separable (MDS) codes form an important family of quantum codes. In this paper, using Hermitian construction and classical constacyclic codes, we construct six classes of quantum MDS codes. Two of these six classes of quantum MDS codes have larger minimum distance than the ones available in the literature. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature.
Quantum maximal-distance-separable (MDS) codes form an important class of
quantum codes. It is very hard to construct quantum MDS codes with relatively
large minimum distance. In this paper, based on classical constacyclic codes,
we construct two classes of quantum MDS codes with parameters
where , and with r even, and
where , and with r odd. The quantum MDS codes
exhibited here have parameters better than the ones available in the
literature.
One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. Quantum maximal distance separable (MDS) codes are optimal in the sense they attain maximal minimum distance. Recently, constructing quantum MDS codes has received much attention and seems to become more and more difficult. In this paper, based on classical constacyclic codes, we construct some new quantum MDS codes by employing the Hermitian construction. Compared with the known quantum MDS codes, these quantum MDS codes have much larger minimum distance.
Let q be an odd prime power. Based on classical negacyclic codes, we construct two classes of quantum maximum-distance-separable (MDS) codes with parameters [[q2+1, q2-2d+3, d]]q where q ≡ 1 (mod 4) and 2 ≤ d ≤ q+1 is even, and [[(q2+1)/2,(q2+1)/2-2d+2,d]]q where 3 ≤ d ≤ q is odd. Some of these quantum MDS codes are new in the sense that their parameters are different from all the previously known ones.
We construct maximum distance separable quantum error-correcting codes. The codes are defined over q-dimensional quantum systems, where q is any prime power. The construction yields quantum MDS codes of length up to q+1 for all possible dimensions and some quantum MDS codes of length up to q2+1. In particular, those families contain codes [[6,2,3]]p and [[7,3,3]]p for pges3 (cf. [K. Feng (2002)])
New quantum maximum-distance-separable (MDS) codes with parameters [[ q 2 + 1, q 2 -2 d + 3, d ]]q, where q =2t, t ≥ 1 and 3 ≤ d ≤ q +1 is an odd integer, are constructed in this paper.
We present several results on quantum codes over general alphabets
(that is, in which the fundamental units may have more than two states).
In particular, we consider codes derived from finite symplectic geometry
assumed to have additional global symmetries. From this standpoint, the
analogs of Calderbank-Shor-Steane codes and of GF(4)-linear codes turn
out to be special cases of the same construction. This allows us to
construct families of quantum codes from certain codes over number
fields; in particular, we get analogs of quadratic residue codes,
including a single-error-correcting code encoding one letter in five,
for any alphabet size. We also consider the problem of fault-tolerant
computation through such codes, generalizing ideas of Gottesman (see
Phys. Rev. A, vol.57, no.1, p127-37, 1998)
On optimal quantum codes
- M Grassl
- T Beth
- M Rottler
M. Grassl, T. Beth, and M. Rottler, On optimal quantum codes, Int. J. Quantum Inf. 2(1), 757-775
(2004).
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- X Zhu
L. Hu, Q. Yue, X. Zhu, New quantum MDS codes from constacyclic codes, Chinese Ann. Math.
Ser B. 37, 891-898 (2016).
- X Fang
- J Luo
X. Fang, J. Luo, New quantum MDS codes over Finite Fields. Quantum Inf. Process. 19(1), 16
(2020).