Math 641 - Introduction to Error-Correcting Codes
- Prerequisites: Math 320 or 340; and Math 541 or consent of instructor.
- Frequency: Irregular.
- Student Body: Math, Comp Sci, Stat and Elec Eng advanced undergraduate and graduate students
- Credits: 3 (N-A)
- Recent Texts: Error-Correcting Codes and Finite Fields (O. Pretzel), Algebraic Codes for Data Transmission (R. Blahut)
- Course Coordinator: Richard A. Brualdi
- Background and Goals: Math/ECE 641 is a first course on coding theory with emphasis on linear codes and covering the basics of the subject. It is essential that the student have a good knowledge of basic linear algebra; some knowledge of abstract algebraic systems (finite fields, polynomial rings and ideals, ...) is very helpful. We shall have to develop some of the theory of finite fields. This is necessary even if one is only interested in binary codes.
- Alternatives: n/a
- Subsequent Courses: n/a
Content coverage:
- Shannon's fundamental theorem on the capacity of a channel
- The quest of coding theorists to realize the Shannon capacity
- Linear codes
- Basic parameters of codes and their relationships
- Perfect codes - Hamming and Golay codes
- Dual codes, weight enumerators and the MacWilliams identities for the dual code
- Reed-Muller codes
- Cyclic codes
- Reed-Solomon codes (used on compact discs) and the Discrete Fourier Transform
- BCH codes
- Bounds on code parameters
- Encoding and decoding techniques
- Goppa codes
In order to construct and investigate codes, we will have to develop finite fields sufficiently to get a good computational understanding of their construction, arithmetic, and algebra.
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