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New Quantum MDS Codes with Flexible Parameters from Hermitian Self-Orthogonal GRS Codes
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Abstract
Let q be a prime power. Let be a divisor of , and let and be divisors of q+1. Under certain conditions we prove that there exists an MDS stabilizer quantum code with length where . This is a flexible construction, which includes new MDS parameters not known before.
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Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In recent years, Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes have been widely used to construct quantum MDS codes. In this paper, we give some sufficient conditions under which a certain system of equations over has a solution over , which effectively unify similar known techniques for constructing Hermitian self-orthogonal codes. Moreover, we construct five new classes of q-ary quantum MDS codes with flexible parameters from Hermitian self-orthogonal GRS codes. Compared to the previous literature, the quantum MDS codes we construct have different lengths, or the same length but larger distances. In particular, some of the quantum MDS codes we construct have distances that can be taken to the maximum distance of the quantum MDS codes from GRS 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.
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.
We construct a family of quantum maximum-distance-separable (MDS) codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them analytically. We find that existing quantum MDS codes can be unified under these codes in the sense that when a quantum MDS code exists, then a quantum code of this type with the same parameters also exists. Thus, as far as is known at present, they are the most important family of quantum MDS codes.
In this paper, we first construct several classes of classical Hermitian self-orthogonal maximum distance separable (MDS) codes. Through these classical codes, we are able to obtain various quantum MDS codes. It turns out that many of our quantum codes are new in the sense that the parameters of our quantum codes cannot be obtained from all previous constructions.
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...
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.
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.
One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over Fq 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose-Chaudhuri-Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper
Choose λ = 2, τ = 3, ρ = q+1
- Proof
Proof. Choose λ = 2, τ = 3, ρ = q+1. Then κ = gcd(λ, ρ)·gcd
Quantum Codes from Generalized Monomial-Cartesian Codes
- B Barbero-Lucas
- F Hernando
- H Martín-Cruz
- G Mcguire
- Hermitian Mds
- Mds Almost
B. Barbero-Lucas, F. Hernando, H. Martín-Cruz, and G. McGuire. MDS, Hermitian
Almost MDS, and Gilbert-Varshamov Quantum Codes from Generalized Monomial-Cartesian Codes. Quantum Inf Process, 23(86), 2024.