all constructions presented here are performed algebraically, i.e., the procedures
adopted are capable of constructing families of codes, and not only codes with
specific parameters.
Description of the Book
The book is organized in such a way that the reader can skip some introductory
chapters without major problems.
Chapter 1 presents a review of some basic concepts on linear algebra and metric
spaces, necessary to define the scenario and the structure of the quantum mechanics.
In Chap. 2, the postulates of quantum mechanics, the definition of single and
multiple qubit gates and the most common types of quantum channels are reviewed.
Chapter 3 is concerned with the first constructions or construction methods of
quantum codes shown in the literature. The well-known five qubit and the Steane
code are examples of such codes. We also review the CSS construction and the
stabilizer quantum code construction.
Chapter 4 is devoted to review some definitions and results on linear block
codes. We recall the concept of Euclidean and Hermitian dual of a linear code as
well as the techniques to obtain new codes from old. Additionally, the classes of
cyclic and algebraic geometry codes are reviewed, since such classical linear codes
are necessary for the quantum code constructions presented in this book.
Chapter 5 brings the most relevant constructions of quantum codes that we have
published in the last ten years of research. They include several families of quantum
codes derived from (classical) Bose–Chaudhuri–Hocquenghem (BCH) and from
(classical) algebraic geometry codes. Moreover, constructions of quantum synchronizable codes derived from (classical) cyclic, BCH and product codes are also
presented here.
In Chap. 6, as in Chap. 5, we present my most relevant contributions concerning
asymmetric quantum code constructions, which were published in the last years.
We construct several families of asymmetric quantum codes (AQQs) derived from
(classical) Reed–Solomon and generalized Reed–Solomon codes, generalized
Reed–Muller and BCH codes. Additionally, we generalize to AQQs the
well-known methods which are valid to quantum codes, namely: puncturing,
extending, expanding, direct sum and the ðuju þ vÞ construction.
In Chap. 7, we present my main contributions published in the last years concerning constructions of quantum convolutional codes. We explain how to construct families of quantum convolutional codes with good parameters derived from
viii
Preface
adopted are capable of constructing families of codes, and not only codes with
specific parameters.
Description of the Book
The book is organized in such a way that the reader can skip some introductory
chapters without major problems.
Chapter 1 presents a review of some basic concepts on linear algebra and metric
spaces, necessary to define the scenario and the structure of the quantum mechanics.
In Chap. 2, the postulates of quantum mechanics, the definition of single and
multiple qubit gates and the most common types of quantum channels are reviewed.
Chapter 3 is concerned with the first constructions or construction methods of
quantum codes shown in the literature. The well-known five qubit and the Steane
code are examples of such codes. We also review the CSS construction and the
stabilizer quantum code construction.
Chapter 4 is devoted to review some definitions and results on linear block
codes. We recall the concept of Euclidean and Hermitian dual of a linear code as
well as the techniques to obtain new codes from old. Additionally, the classes of
cyclic and algebraic geometry codes are reviewed, since such classical linear codes
are necessary for the quantum code constructions presented in this book.
Chapter 5 brings the most relevant constructions of quantum codes that we have
published in the last ten years of research. They include several families of quantum
codes derived from (classical) Bose–Chaudhuri–Hocquenghem (BCH) and from
(classical) algebraic geometry codes. Moreover, constructions of quantum synchronizable codes derived from (classical) cyclic, BCH and product codes are also
presented here.
In Chap. 6, as in Chap. 5, we present my most relevant contributions concerning
asymmetric quantum code constructions, which were published in the last years.
We construct several families of asymmetric quantum codes (AQQs) derived from
(classical) Reed–Solomon and generalized Reed–Solomon codes, generalized
Reed–Muller and BCH codes. Additionally, we generalize to AQQs the
well-known methods which are valid to quantum codes, namely: puncturing,
extending, expanding, direct sum and the ðuju þ vÞ construction.
In Chap. 7, we present my main contributions published in the last years concerning constructions of quantum convolutional codes. We explain how to construct families of quantum convolutional codes with good parameters derived from
viii
Preface