scholarly journals Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1495
Author(s):  
Yansheng Wu ◽  
Yoonjin Lee
Keyword(s):  
Linear Codes ◽  
Sufficient Conditions ◽  
Quantum Codes ◽  
Binary Linear Codes ◽  
Hamming Codes ◽  
Orthogonal Codes ◽  
Css Codes ◽  
Css Construction ◽  
Necessary And Sufficient ◽  
Quantum Error

It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.

Quantum codes over rings

2014 ◽  
Vol 12 (04) ◽  
pp. 1450020 ◽  
Author(s):  
Kenza Guenda ◽  
T. Aaron Gulliver
Keyword(s):  
Linear Codes ◽  
Error Correcting Codes ◽  
Codes Over Rings ◽  
Quantum Codes ◽  
Matrix Product ◽  
Frobenius Rings ◽  
Linear Codes Over Rings ◽  
Generalized Gray Map ◽  
Css Construction ◽  
Quantum Error

This paper considers the construction of quantum error correcting codes from linear codes over finite commutative Frobenius rings. We extend the Calderbank–Shor–Steane (CSS) construction to these rings. Further, quantum codes are extended to matrix product codes. Quantum codes over 𝔽pk are also obtained from linear codes over rings using the generalized Gray map.

scholarly journals Reinterpreting the Stabilizer Formalism

2021 ◽  
Author(s):  
Vatsal Pramod Jha ◽  
Udaya Parampalli ◽  
Abhay Kumar Singh
Keyword(s):  
Binary Codes ◽  
Quantum Codes ◽  
Stabilizer Codes ◽  
Orthogonal Codes ◽  
Css Construction

<div>Stabilizer codes, introduced in [2], [3], have been a prominent example of quantum codes constructed via classical codes. The paper [3], introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we reinterpret the stabilizer formalism by considering binary codes over the symbol-pair metric (see [9]). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters.</div>

On quantum codes obtained from cyclic codes over A2

2015 ◽  
Vol 13 (03) ◽  
pp. 1550031 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Gray Map ◽  
Quantum Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Length ◽  
Necessary And Sufficient ◽  
Quantum Error

In this paper, quantum codes from cyclic codes over A2 = F2 + uF2 + vF2 + uvF2, u2 = u, v2 = v, uv = vu, for arbitrary length n have been constructed. It is shown that if C is self orthogonal over A2, then so is Ψ(C), where Ψ is a Gray map. A necessary and sufficient condition for cyclic codes over A2 that contains its dual has also been given. Finally, the parameters of quantum error correcting codes are obtained from cyclic codes over A2.

Quantum codes from a class of constacyclic codes over finite commutative rings

2019 ◽  
Vol 19 (12) ◽  
pp. 2150003 ◽  
Author(s):  
Hai Q. Dinh ◽  
Tushar Bag ◽  
Ashish K. Upadhyay ◽  
Mohammad Ashraf ◽  
Ghulam Mohammad ◽  
...  
Keyword(s):  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Negacyclic Codes ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Necessary And Sufficient

Let [Formula: see text] be an odd prime, and [Formula: see text] be an integer such that [Formula: see text]. Using pairwise orthogonal idempotents [Formula: see text] of the ring [Formula: see text], with [Formula: see text], [Formula: see text] is decomposed as [Formula: see text], which contains the ring [Formula: see text] as a subring. It is shown that, for [Formula: see text], [Formula: see text], and it is invertible if and only if [Formula: see text] and [Formula: see text] are units of [Formula: see text]. In such cases, we study [Formula: see text]-constacyclic codes over [Formula: see text]. We present a direct sum decomposition of [Formula: see text]-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a [Formula: see text]-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over [Formula: see text], with better parameters than existing ones, are constructed from cyclic and negacyclic codes over [Formula: see text].

Some results on the linear codes over the finite ring F2+v1F2+⋯+vrF2

2016 ◽  
Vol 14 (01) ◽  
pp. 1650012 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Finite Ring ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Quantum Error ◽  
Quasi Cyclic Codes

In this paper, we study the structure of cyclic, quasi-cyclic codes and their skew codes over the finite ring [Formula: see text], [Formula: see text] for [Formula: see text]. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic codes over [Formula: see text] are obtained. A necessary and sufficient condition for cyclic code over [Formula: see text] that contains its dual has been given. The parameters of quantum error correcting codes are obtained from cyclic codes over [Formula: see text].

scholarly journals Reinterpreting the Stabilizer Formalism

2021 ◽  
Author(s):  
Vatsal Pramod Jha ◽  
Udaya Parampalli ◽  
Abhay Kumar Singh
Keyword(s):  
Binary Codes ◽  
Quantum Codes ◽  
Stabilizer Codes ◽  
Orthogonal Codes ◽  
Css Construction

<div>Stabilizer codes, introduced in [2], [3], have been a prominent example of quantum codes constructed via classical codes. The paper [3], introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we reinterpret the stabilizer formalism by considering binary codes over the symbol-pair metric (see [9]). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters.</div>

scholarly journals Some New Quantum BCH Codes over Finite Fields

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 712
Author(s):  
Lijuan Xing ◽  
Zhuo Li
Keyword(s):  
Finite Fields ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Quantum Codes ◽  
Dual Codes ◽  
Cyclotomic Cosets ◽  
New Family ◽  
Consecutive Integers ◽  
Css Construction ◽  
Quantum Error

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose–Chaudhuri–Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane’s enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.

New constructions of entanglement-assisted quantum MDS codes from negacyclic codes

2019 ◽  
Vol 17 (03) ◽  
pp. 1950022 ◽  
Author(s):  
Ruihu Li ◽  
Guanmin Guo ◽  
Hao Song ◽  
Yang Liu
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Error Correcting Codes ◽  
Optimal Number ◽  
Mds Codes ◽  
Quantum Codes ◽  
General Statement ◽  
Negacyclic Codes ◽  
Maximum Distance Separable ◽  
Quantum Error

When constructing quantum codes under the entanglement-assisted (EA) stabilizer formalism, one can ignore the limitation of dual-containing condition. This allows us to construct EA quantum error-correcting codes (QECCs) from any classical linear codes. The main contribution of this manuscript is to make a general statement for determining the optimal number of pre-shared qubits instead of presenting only specific cases. Let [Formula: see text] and [Formula: see text], where [Formula: see text] is an odd prime power, [Formula: see text] and [Formula: see text]. By deeply investigating the decomposition of the defining set of negacyclic codes, we generalize the number of pre-shared entanglement pairs of Construction (1) in Lu et al. [Quantom Inf. Process. 17 (2018) 69] from [Formula: see text] to arbitrary even numbers less than or equal to [Formula: see text]. Consequently, a series of EA quantum maximum distance separable (EAQMDS) codes can be produced. The absolute majority of them are new and the minimum distance can be up to [Formula: see text]. Moreover, this method can be applied to construct many other families of EAQECCs with good parameters, especially large minimum distance.

Constacyclic codes over the ring Fq[u,v,w]/〈u2 − 1, v2 − 1,w3 − w, uv − vu,vw − wv,wu − uw〉

2021 ◽  
Author(s):  
Shunhua Zhang
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Gray Map ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Direct Sum ◽  
Necessary And Sufficient

Let [Formula: see text] be the ring [Formula: see text], where [Formula: see text] for any odd prime [Formula: see text] and positive integer [Formula: see text]. In this paper, we study constacyclic codes over the ring [Formula: see text]. We define a Gray map by a matrix and decompose a constacyclic code over the ring [Formula: see text] as the direct sum of constacyclic codes over [Formula: see text], we also characterize self-dual constacyclic codes over the ring [Formula: see text] and give necessary and sufficient conditions for constacyclic codes to be dual-containing. As an application, we give a method to construct quantum codes from dual-containing constacyclic codes over the ring [Formula: see text].

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