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Classical self-orthogonal codes and their applications to quantum codes

Lingfei Jin

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

After the pioneering work of Shor and Steane, we are able to establish links between quantum codes and classical codes with certain self-orthogonality. Therefore, constructing classical self-orthogonal codes with small dimension and large dual distance becomes our centre point due to the interesting application to quantum codes. \n \n \n \nOne of the most interesting and useful families of classical codes are MDS codes. We will show a systematic construction for classical Hermitian self-orthogonal MDS codes through generalized Reed-Solomn codes. Afterwards, new families of quantum MDS codes can be produced. \n \n \n \nClassical BCH codes would be another good choice for self-orthogonal codes. We show the dual containment through polynomial evaluations, simply by choosing suitable cyclotomic cosets. As a result, quantum codes with good parameters can be derived. \n \n \n \nAnother good candidate of classical codes to investigate for dual containment comes from the class of \n \nalgebraic geometry codes. Instead of testing for dual containment directly, this thesis explores an equivalent condition for the existence of self-orthogonal AG codes. \n \n \n \nAfter various constructions for quantum codes, this thesis ends with a study on bounds for estimating the parameters of codes.

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After the pioneering work of Shor and Steane, we are able to establish links between quantum codes and classical codes with certain self-orthogonality. Therefore, constructing classical self-orthogonal codes with small dimension and large dual distance becomes our centre point due to the interesting application to quantum codes. \n \n \n \nOne of the most interesting and useful families of classical codes are MDS codes. We will show a systematic construction for classical Hermitian self-orthogonal MDS codes through generalized Reed-Solomn codes. Afterwards, new families of quantum MDS codes can be produced. \n \n \n \nClassical BCH codes would be another good choice for self-orthogonal codes. We show the dual containment through polynomial evaluations, simply by choosing suitable cyclotomic cosets. As a result, quantum codes with good parameters can be derived. \n \n \n \nAnother good candidate of classical codes to investigate for dual containment comes from the class of \n \nalgebraic geometry codes. Instead of testing for dual containment directly, this thesis explores an equivalent condition for the existence of self-orthogonal AG codes. \n \n \n \nAfter various constructions for quantum codes, this thesis ends with a study on bounds for estimating the parameters of codes.
A new criterion for the existence of Hermitian dual-containing cyclic codes is obtained based on a characterization of q-cyclotomic cosets modulo n. This criterion admits a method of constructing new Hermitian dual-containing cyclic codes (hence new quantum codes) from the old ones, and enables us to construct several classes of quantum maximum-distance-separable (MDS) codes from classical Hermitian dual-containing cyclic codes. We analyse that our codes are good in certain sense. Besides, some enumerative conclusions are also presented as a result of the characterization of q-cyclotomic cosets modulo n.
In this paper, two classes of quantum MDS codes are constructed. The main tools are multiplicative structures on finite fields. Carefully choosing different cosets can make the corresponding generalized Reed-Solomon codes Hermitian self-orthogonal, which yields quantum MDS codes by utilizing Hermitian type CSS construction. In some cases, these codes have minimal distances larger than q/2q/2q/2. Also, the parameters of these codes have never been reported before.
It is conjectured that quantum computers are able to solve certain problems more \nquickly than any deterministic or probabilistic computer. For instance, Shor's algorithm \nis able to factor large integers in polynomial time on a quantum computer. \nA quantum computer exploits the rules of quantum mechanics to speed up computations. \nHowever, it is a formidable task to build a quantum computer, since the \nquantum mechanical systems storing the information unavoidably interact with their \nenvironment. Therefore, one has to mitigate the resulting noise and decoherence \neffects to avoid computational errors. \nIn this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the \nfundamental theory and necessary background of quantum codes and construct many \nfamilies of quantum block and convolutional codes over finite fields, in addition to \nfamilies of subsystem codes. This dissertation is organized into three parts: \nQuantum Block Codes. After introducing the theory of quantum block codes, I \nestablish conditions when BCH codes are self-orthogonal (or dual-containing) \nwith respect to Euclidean and Hermitian inner products. In particular, I derive \ntwo families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum \ncodes, as well as families of quantum codes derived from projective geometries. \nSubsystem Codes. Subsystem codes form a new class of quantum codes in which \nthe underlying classical codes do not need to be self-orthogonal. I give an \nintroduction to subsystem codes and present several methods for subsystem \ncode constructions. I derive families of subsystem codes from classical BCH and \nRS codes and establish a family of optimal MDS subsystem codes. I establish \npropagation rules of subsystem codes and construct tables of upper and lower \nbounds on subsystem code parameters. \nQuantum Convolutional Codes. Quantum convolutional codes are particularly \nwell-suited for communication applications. I develop the theory of quantum \nconvolutional codes and give families of quantum convolutional codes based \non RS codes. Furthermore, I establish a bound on the code parameters of \nquantum convolutional codes - the generalized Singleton bound. I develop a \ngeneral framework for deriving convolutional codes from block codes and use it \nto derive families of non-catastrophic quantum convolutional codes from BCH \ncodes. \nThe dissertation concludes with a discussion of some open problems.
The construction of quantum maximum-distance-separable (MDS) codes has become one of the major goals in quantum coding theory. In this paper, we construct several classes of Hermitian self-orthogonal generalized Reed–Solomon codes. Based on these classical MDS codes, we obtain several new classes of quantum MDS codes with large minimum distance. It turns out that these constructed quantum MDS codes have fewer constraints on the selection of code length, and in some cases, have larger minimum distance than previous literature. Notably, some of the distance parameters of our codes are greater than (q/2)+1.