quantumerror correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantumerror correction is that it is significantly more difficult to design ...
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quantumerror correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantumerror correction is that it is significantly more difficult to design error-correctingcodes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantumerror-correctingcodes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum information science, where in principle any binary linear code can be turned into a quantumerror-correctingcode by assuming a small number of reliable quantum bits. This paper studies how best to take advantage of these latest ideas to construct desirable quantumerror-correctingcodes of very high information rate. Our methods exploit structured high-rate low-density parity-check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths. Our approach to designing high-rate quantumerror-correctingcodes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation where promising quantum analogues of low-density parity-check codes are difficult to find.
The Euclidean hull dimension of a linear code is an important quantity to determine the parameters of an entanglement-assisted quantum error-correcting code (EAQECC) if the Euclidean construction is applied. In this p...
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The Euclidean hull dimension of a linear code is an important quantity to determine the parameters of an entanglement-assisted quantum error-correcting code (EAQECC) if the Euclidean construction is applied. In this paper, we study the Euclidean hull of a linear code by means of orthogonal matrices. We provide some methods to construct linear codes over F-pm with hull of arbitrary dimensions. With existence of self-dual bases of F-pm over F-p, we determine a Gray map from F-pm to F-p(m), and from a given linear code over F-pm with one-dimensional hull, we construct, using such a Gray map, a linear code over F-p with m-dimensional hull for all m when p is even and for all m odd when p is odd. Comparisons with classical constructions are made, and some good EAQECCs over F-q, q = 2, 3, 4, 5, 9, 13, 17, 49 are presented.
It is possible to construct an entanglement-assistedquantumerror-correcting (EAQEC, for short) code from any classical linear code. However, the parameter of ebits is usually calculated by computer search. In this w...
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It is possible to construct an entanglement-assistedquantumerror-correcting (EAQEC, for short) code from any classical linear code. However, the parameter of ebits is usually calculated by computer search. In this work, we can construct a family of EAQEC codes from arbitrary binary linear codes, where the parameter of ebits can be easily generated algebraically and not by computational search. Moreover, the constructed EAQEC codes are maximal-entanglement EAQEC codes. We also present a different method of constructing entanglement-assisted accumulator codes. Finally, we prove that asymptotically good EAQEC codes exist.
The Euclidean hull of a linear code C is the intersection of C with its Euclidean dual C-perpendicular to. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algor...
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The Euclidean hull of a linear code C is the intersection of C with its Euclidean dual C-perpendicular to. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correctingcodes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented.
The hull of a linear code C is the intersection of C with its dual C-perpendicular to. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing ...
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The hull of a linear code C is the intersection of C with its dual C-perpendicular to. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The hull of linear codes has recently found its application to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, we provide a new method to construct linear codes with one-dimensional hull. This construction method improves the code lengths and dimensions of the recent results given by the author. As a consequence, we derive several new classes of optimal linear codes with one-dimensional hull. Some new EAQECCs are presented.
This paper investigates symplectic hulls of linear codes. We use a different view to obtain more structural properties of generator matrices with respect to the symplectic inner product. As an outgrowth, generalized f...
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This paper investigates symplectic hulls of linear codes. We use a different view to obtain more structural properties of generator matrices with respect to the symplectic inner product. As an outgrowth, generalized formulas for calculating dimensions of symplectic hulls are derived, which extend some known results in the literature. We then study the symplectic hull-variation problem and prove that a monomially equivalent linear code with a smaller dimensional symplectic hull can always be explicitly derived from a given q-ary symplectic self-dual code with a standard generator matrix for q >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ge 3$$\end{document}. As an application, we present an improved propagation rule for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain some new and record-breaking binary EAQECCs.
We construct a lot of optimal or near-optimal [n, k, d](9 )Hermitian self-orthogonal codes for k <= 3 using norm codes and matrix combinatorial construction method. As an application, we construct nine families of ...
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We construct a lot of optimal or near-optimal [n, k, d](9 )Hermitian self-orthogonal codes for k <= 3 using norm codes and matrix combinatorial construction method. As an application, we construct nine families of entanglement-assisted quantum error-correcting codes. Some of these codes can achieve q-ary linear EA-Griesmer bound with better parameters than those in the literature
entanglement-assisted quantum error-correcting codes are not only theoretically interesting, but also of great importance in practical physically applications. In this work, two class of entanglement-assistedquantum ...
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ISBN:
(纸本)9781509023776
entanglement-assisted quantum error-correcting codes are not only theoretically interesting, but also of great importance in practical physically applications. In this work, two class of entanglement-assistedquantumcodes are constructed. The first class of entanglement-assistedquantumcodes is minimal ebits entanglement-assistedquantumcodes, which require only one copy of maximally entangled state no matter how large the code length is. The second class of entanglement-assistedquantumcodes is maximal entanglemententanglement-assistedquantumcodes, which can achieve the entanglement-assisted hashing bound asymptotically.
In this work, we first generalize the sigma-LCD codes over finite fields to sigma-LCD codes over finite chain rings. Under suitable conditions, linear codes over finite chain rings that are sigma-LCD codes are charact...
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In this work, we first generalize the sigma-LCD codes over finite fields to sigma-LCD codes over finite chain rings. Under suitable conditions, linear codes over finite chain rings that are sigma-LCD codes are characterized. Then we provide a necessary and sufficient condition for free constacyclic codes over finite chain rings to be sigma-LCD. We also get some new binary LCD codes of different lengths which come from Gray images of constacyclic sigma-LCD codes over F-2+ gamma F-2+ gamma F-2(2). Finally, for special finite chain rings F-q + gamma F-q, we define a new Gray map Phi from (F-q + gamma F-q)(n) to F-q(2n), and by using sigma--LCD codes over finite chain rings F-q + gamma F-q, we construct new entanglement-assistedquantumerror-correcting (abbreviated to EAQEC) codes with maximal entanglement and parts of them are MDS EAQEC codes.
entanglement-assisted quantum error-correcting codes as a generalization of stabilizer quantumerror-correcting (QEC) codes can improve the performance of stabilizer QEC codes and can be constructed from arbitrary cla...
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entanglement-assisted quantum error-correcting codes as a generalization of stabilizer quantumerror-correcting (QEC) codes can improve the performance of stabilizer QEC codes and can be constructed from arbitrary classical linear codes by relaxing the dual-containing condition and using pre-shared entanglement states between the sender and the receiver. In this paper, we construct some families of entanglement-assistedquantum maximum distance separable codes with parameters [[q(2)-1/a, q(2)-1/a-2(d - 1)+ c, d;c]](q), where q is an odd prime power with the form q = am +/- l, a = l(2) - 1 or a = l(2)-1/2, l is an odd integer, and m is a positive integer. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.
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