# Coding Theory

Math 625
Course number 21044
Spring 2006
Meeting: TTh, 4:00- 5:15 pm.
GMCS 325
San Diego State University
Final Exam: Tues. May 16, 3:30-5:30.

Professor: Mike O'Sullivan
Email: m.osullivan@math.sdsu.edu
OfficeGMCS 579, ext. 594-6697
Office Hours: MTWTh 2:30-3:30, W 5:15-6:00. (subject to change)
Other times: by appointment.

## Course Description

This course concerns error-control coding, the effort to correct errors that occur in the transmission and storage of data. We will focus on three types of codes and the algorithms that are used to decode them: Reed-Solomon codes and the Berlekamp-Massey algorithm, convolutional codes and the Viterbi algorithm, low-density parity-check codes and sum-product algorithm. The first two codes are ubiquitous in modern communications systems, while low-density parity-check codes are a promising alternative that has dominated research conferences in the last few years. All three codes were developed in the late 1950's and early 1960's, but low-density parity-check codes were not appreciated until the last decade.

These three families of codes lead to a great variety of interesting mathematics. The construction and algorithms for encoding and decoding Reed-Solomon codes employ the algebraic theory of finite fields and polynomial rings. The code vectors take values in a finite field and an error switches one field element to another. The Berlekamp-Massey algorithm computes a polynomial that identifies where the error occured.

There is also a beautiful algebraic theory for convolutional codes, but in engineering practice the algebraic aspects are minimized. Convolutional codes are usually binary and they are modeled by circuit diagrams. Errors are probabilistic: the decoder receives a probability that a given bit is a 0 or 1. The Viterbi algorithm then attempts to find the most likely codeword given the input probabilities.

Low-density parity check codes are also studied primarily in the binary case. There is a highly efficient decoding algorithm for these codes, based on probabilistic reasoning, that is similar to the Viterbi algorithm and to algorithms used in artificial intelligence. Decoding can be modeled by messages, passed along the edges of a bipartite graph, that represent ``beliefs'' about the binary values of a codeword. Research into LDPC codes is very active and includes analysis of the decoding algorithm, both experimental and theoretical, and construction of good codes using a variety of methods, random, algebraic, geometric, and combinatorial.

## Course Materials

No textbooks cover exactly what I want, so the course will be based on my lectures and course notes. A list of references and recommendations for articles and book chapters will be given as the course develops. Computer experimentation will be required. You may use Maple, Matlab or Magma or another system if you'd like. Mytutorials for Maple and Magma may be of some help.

Here is the first installment of the notes Polynomials and Finite Fields