- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Coding theory and algebraic curves
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Coding theory and algebraic curves Edwards, Brandon Gary
Abstract
In 1981 V.D. Goppa [9] used the evaluation of rational functions on algebraic curves to define a new and exciting class of error correcting codes now known as geometric Goppa codes. As with any other error correcting code however, this construction procedure alone was only the beginning of what was need in order that these codes could be used in practice. Procedures needed to be found to aid in both the explicit construction of certain geometric Goppa codes and the creation of efficient decoding algorithms. The problems surrounding these concerns have since motivated many coding theorists to become actively involved in the study of algebraic curves and have equally sparked the interests of algebraic geometers. In this paper I introduce the topic of coding theory and highlight the successes and failures that have occurred in the attempt to bring geometric Goppa codes to their full stage of implementation. In the first two chapters, I give a basic introduction to the theory of error correcting codes, assuming no previous knowledge of the subject. Chapter 3 is concerned with the construction of rational geometric Goppa codes for which no knowledge of algebraic curves is necessary. Various positive and negative aspects of these codes are discussed which motivates the introduction of the general class of geometric Goppa codes in the following two chapters. For the material in Chapters 4 and 5, I will assume that the reader is familiar with divisors and linear systems on non-singular projective curves. After I define the class of geometric Goppa codes, I go on to discuss the advances that have been made in the search for explicit constructions of some of the best codes in this class. Finally, I present some basic decoding algorithms for both rational and general geometric Goppa codes. These algorithms demonstrate how easy it is to develop efficient decoders in both of these cases.
Item Metadata
| Title |
Coding theory and algebraic curves
|
| Creator | |
| Publisher |
University of British Columbia
|
| Date Issued |
1997
|
| Description |
In 1981 V.D. Goppa [9] used the evaluation of rational functions on algebraic curves to define
a new and exciting class of error correcting codes now known as geometric Goppa codes. As
with any other error correcting code however, this construction procedure alone was only the
beginning of what was need in order that these codes could be used in practice. Procedures
needed to be found to aid in both the explicit construction of certain geometric Goppa codes and
the creation of efficient decoding algorithms. The problems surrounding these concerns have
since motivated many coding theorists to become actively involved in the study of algebraic
curves and have equally sparked the interests of algebraic geometers.
In this paper I introduce the topic of coding theory and highlight the successes and failures that
have occurred in the attempt to bring geometric Goppa codes to their full stage of implementation.
In the first two chapters, I give a basic introduction to the theory of error correcting codes,
assuming no previous knowledge of the subject. Chapter 3 is concerned with the construction
of rational geometric Goppa codes for which no knowledge of algebraic curves is necessary. Various
positive and negative aspects of these codes are discussed which motivates the introduction
of the general class of geometric Goppa codes in the following two chapters. For the material
in Chapters 4 and 5, I will assume that the reader is familiar with divisors and linear systems
on non-singular projective curves. After I define the class of geometric Goppa codes, I go on
to discuss the advances that have been made in the search for explicit constructions of some of
the best codes in this class. Finally, I present some basic decoding algorithms for both rational
and general geometric Goppa codes. These algorithms demonstrate how easy it is to develop
efficient decoders in both of these cases.
|
| Extent |
1689103 bytes
|
| Genre | |
| Type | |
| File Format |
application/pdf
|
| Language |
eng
|
| Date Available |
2009-03-12
|
| Provider |
Vancouver : University of British Columbia Library
|
| Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
|
| DOI |
10.14288/1.0079766
|
| URI | |
| Degree | |
| Program | |
| Affiliation | |
| Degree Grantor |
University of British Columbia
|
| Graduation Date |
1997-05
|
| Campus | |
| Scholarly Level |
Graduate
|
| Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.