Since Calderbank et al. established CRSS construction, which constructs quantum error-correcting codes (QECC) from classical error-correcting codes, many QECC demonstrating good performance have been constructed from classical error-correcting codes; these codes are referred to as quantum stabilizer codes. Recent physical experiments indicate that the probabilities of occurrence of qubit- ip errors are much less than those of phase- ip errors in most quantum mechanical systems. Error-correcting codes thus constructed are referred to as asymmetric quantum codes. In this paper, series of asymmetric quantum stabilizer codes are constructed using nested Goppa codes and dual contained Goppa codes. In the construction of asymmetric quantum codes based on nested Goppa codes, the choice of Goppa codes is limited to construct codes with analysis methods. In normal circumstances, the construction is executed with the help of mathematical software Matlab, which is used to compute the minimum distance of Goppa codes. In the construction of asymmetric quantum codes based on dual contained Goppa codes, the error correction ability of the codes constructed in this paper is re ected in the correction of Z-type errors
Since Calderbank et al. established CRSS construction, which constructs quantum error-correcting codes (QECC) from classical error-correcting codes, many QECC demonstrating good performance have been constructed from classical error-correcting codes; these codes are referred to as quantum stabilizer codes. Recent physical experiments indicate that the probabilities of occurrence of qubit- ip errors are much less than those of phase- ip errors in most quantum mechanical systems. Error-correcting codes thus constructed are referred to as asymmetric quantum codes. In this paper, series of asymmetric quantum stabilizer codes are constructed using nested Goppa codes and dual contained Goppa codes. In the construction of asymmetric quantum codes based on nested Goppa codes, the choice of Goppa codes is limited to construct codes with analysis methods. In normal circumstances, the construction is executed with the help of mathematical software Matlab, which is used to compute the minimum distance of Goppa codes. In the construction of asymmetric quantum codes based on dual contained Goppa codes, the error correction ability of the codes constructed in this paper is re ected in the correction of Z-type errors
Since Calderbank et al.established CRSS construction,which constructs quantum error-correcting codes(QECC) from classical error-correcting codes,many QECC demonstrating good performance have been constructed from classical error-correcting codes;these codes are referred to as quantum stabilizer codes.Recent physical experiments indicate that the probabilities of occurrence of qubit-flip errors are much less than those of phase-flip errors in most quantum mechanical systems.Error-correcting codes thus constructed are referred to as asymmetric quantum codes.In this paper,series of asymmetric quantum stabilizer codes are constructed using nested Goppa codes and dual contained Goppa codes.In the construction of asymmetric quantum codes based on nested Goppa codes,the choice of Goppa codes is limited to construct codes with analysis methods.In normal circumstances,the construction is executed with the help of mathematical software Matlab,which is used to compute the minimum distance of Goppa codes.In the construction of asymmetric quantum codes based on dual contained Goppa codes,the error correction ability of the codes constructed in this paper is reflected in the correction of Z-type errors.
Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit- and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank–Shor–Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose–Chaudhuri–Hocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.
This thesis focuses on the construction and analysis of the parameters of asymmetric quantum error-correcting codes (AQECCs). We use the so-called functional approach to connect AQECCs to classical codes via orthogonal arrays. Based on the functional approach, we derive pure AQECCs from CSS-like constructions that include the standard CSS construction known prior to this work. The CSS-like constructions remove both the specific choice of Euclidean inner product and the $\F_{q}$-linearity condition imposed by the standard variant. Hence, pure AQECCs can now be constructed by using pairs of nested $\F_{r}$-linear codes over $\F_{q}$ where $\F_{r}$ is any subfield of $\F_{q}$ under the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. Relationships between the various CSS-like constructions are also exhibited. A formal definition of asymmetric stabilizer codes is given and a connection between pure asymmetric stabilizer codes and AQECCs derived from the CSS-like constructions is established. It is shown that the class of pure CSS-like AQECCs forms a subset of asymmetric stabilizer codes. We show how classical linear MDS codes can be used to construct AQECCs that satisfy the quantum Singleton bound with dz≥dx≥2 for all possible values of length n for which linear MDS codes over $\F_{q}$ are known to exist. Beyond the lengths specified by the classical MDS conjecture, various explicit constructions of nested pairs of classical codes can be combined with linear programming to establish the optimality of pure CSS-like codes with parameters [[n,k,dz/dx]]q. For q={2,3,4,5,7,8,9}, lists of optimal asymmetric CSS-like codes for reasonable lengths are presented.
This paper characterizes Goppa codes of certain maximal curves over finite fields defined by equations of the form [Formula: see text]. We investigate algebraic geometric and quantum stabilizer codes associated with these maximal curves and propose modifications to improve their parameters. The theoretical analysis is complemented by extensive simulation results, which validate the performance of these codes under various error rates. We provide concrete examples of the constructed codes, comparing them with known results to highlight their strengths and trade-offs. The simulation data, presented through detailed graphs and tables, offers insights into the practical behavior of these codes in noisy environments. Our findings demonstrate that while the constructed codes may not always achieve optimal minimum distances, they offer systematic construction methods and interesting parameter trade-offs that could be valuable in specific applications or for further theoretical study.
Quantum error correction codes (QECCs) can be constructed from the known classical coding paradigm by exploiting the inherent isomorphism between the classical and quantum regimes, while also addressing the challenges imposed by the strange laws of quantum physics. In this spirit, this paper provides deep insights into the duality of quantum and classical coding theory, hence aiming for bridging the gap between them. Explicitly, we survey the rich history of both classical as well as quantum codes. We then provide a comprehensive slow-paced tutorial for constructing stabilizer-based QECCs from arbitrary binary as well as quaternary codes, as exemplified by the dual-containing and non-dual-containing Calderbank-Shor-Steane (CSS) codes, non-CSS codes and entanglement-assisted codes. Finally, we apply our discussions to two popular code families, namely to the family of Bose-Chaudhuri-Hocquenghem as well as of convolutional codes and provide detailed design examples for both their classical as well as their quantum versions.
In this paper, we construct asymmetric quantum error-correcting codes(AQCs) based on subclasses of Alternant codes. Firstly, We propose a new subclass of Alternant codes which can attain the classical Gilbert-Varshamov bound to construct AQCs. It is shown that when dx= 2, Z-parts of the AQCs can attain the classical Gilbert-Varshamov bound. Then we construct AQCs based on a famous subclass of Alternant codes called Goppa codes. As an illustrative example, we get three [[55, 6, 19/4]], [[55, 10, 19/3]], [[55, 15, 19/2]] AQCs from the well known [55, 16, 19] binary Goppa code. At last, we get asymptotically good binary expansions of asymmetric quantum GRS codes, which are quantum generalizations of Retter's classical results. All the AQCs constructed in this paper are pure.