10 research outputs found
The Weight Hierarchies of Linear Codes from Simplicial Complexes
The study of the generalized Hamming weight of linear codes is a significant
research topic in coding theory as it conveys the structural information of the
codes and determines their performance in various applications. However,
determining the generalized Hamming weights of linear codes, especially the
weight hierarchy, is generally challenging. In this paper, we investigate the
generalized Hamming weights of a class of linear code \C over \bF_q, which
is constructed from defining sets. These defining sets are either special
simplicial complexes or their complements in \bF_q^m. We determine the
complete weight hierarchies of these codes by analyzing the maximum or minimum
intersection of certain simplicial complexes and all -dimensional subspaces
of \bF_q^m, where 1\leq r\leq {\rm dim}_{\bF_q}(\C)
Generalized Bezout's Theorem and its applications in coding theory
This paper presents a generalized Bezout theorem which can be used to determine a tighter lower bound of the number of distinct points of intersection of two or more curves for a large class of plane curves. A new approach to determine a lower bound on the minimum distance (and also the generalized Hamming weights) for algebraic-geometric codes defined from a class of plane curves is introduced, based on the generalized Bezout theorem. Examples of more efficient linear codes are constructed using the generalized Bezout theorem and the new approach. For d = 4, the linear codes constructed by the new construction are better than or equal to the known linear codes. For d greater than 5, these new codes are better than the known codes. The Klein code over GF(2(sup 3)) is also constructed
Quantum stabilizer codes and beyond
The importance of quantum error correction in paving the way to build a
practical quantum computer is no longer in doubt. This dissertation makes a
threefold contribution to the mathematical theory of quantum error-correcting
codes. Firstly, it extends the framework of an important class of quantum codes
-- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes
to classical codes over quadratic extension fields, provides many new
constructions of quantum codes, and develops further the theory of optimal
quantum codes and punctured quantum codes. Secondly, it contributes to the
theory of operator quantum error correcting codes also called as subsystem
codes. These codes are expected to have efficient error recovery schemes than
stabilizer codes. This dissertation develops a framework for study and analysis
of subsystem codes using character theoretic methods. In particular, this work
establishes a close link between subsystem codes and classical codes showing
that the subsystem codes can be constructed from arbitrary classical codes.
Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum
codes and considers more realistic channels than the commonly studied
depolarizing channel. It gives systematic constructions of asymmetric quantum
stabilizer codes that exploit the asymmetry of errors in certain quantum
channels.Comment: Ph.D. Dissertation, Texas A&M University, 200
Quantum error control codes
It is conjectured that quantum computers are able to solve certain problems more
quickly than any deterministic or probabilistic computer. For instance, Shor's algorithm
is able to factor large integers in polynomial time on a quantum computer.
A quantum computer exploits the rules of quantum mechanics to speed up computations.
However, it is a formidable task to build a quantum computer, since the
quantum mechanical systems storing the information unavoidably interact with their
environment. Therefore, one has to mitigate the resulting noise and decoherence
effects to avoid computational errors.
In this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the
fundamental theory and necessary background of quantum codes and construct many
families of quantum block and convolutional codes over finite fields, in addition to
families of subsystem codes. This dissertation is organized into three parts:
Quantum Block Codes. After introducing the theory of quantum block codes, I
establish conditions when BCH codes are self-orthogonal (or dual-containing)
with respect to Euclidean and Hermitian inner products. In particular, I derive
two families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum
codes, as well as families of quantum codes derived from projective geometries.
Subsystem Codes. Subsystem codes form a new class of quantum codes in which
the underlying classical codes do not need to be self-orthogonal. I give an
introduction to subsystem codes and present several methods for subsystem
code constructions. I derive families of subsystem codes from classical BCH and
RS codes and establish a family of optimal MDS subsystem codes. I establish
propagation rules of subsystem codes and construct tables of upper and lower
bounds on subsystem code parameters.
Quantum Convolutional Codes. Quantum convolutional codes are particularly
well-suited for communication applications. I develop the theory of quantum
convolutional codes and give families of quantum convolutional codes based
on RS codes. Furthermore, I establish a bound on the code parameters of
quantum convolutional codes - the generalized Singleton bound. I develop a
general framework for deriving convolutional codes from block codes and use it
to derive families of non-catastrophic quantum convolutional codes from BCH
codes.
The dissertation concludes with a discussion of some open problems