MAT 5183

Introduction to algebraic codes.

Sample textbook

[1] Tzuong-Tsien Moh, Introduction to Algebraic Codes, 2008. Freely available to UTSA students.

Catalog entry

Prerequisite: Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.

Contents: (1) Finite Fields: Fields, Polynomial rings, Structure of Finite Fields, Minimal Poly- nomials. (2) Linear codes: Linear codes, Hamming weight and distances, Dual codes, Genera- tor and Parity-Check Matrices (3) Equivalence of Linear codes: Permutation Equivalent Codes, Monomial Equivalent Codes (4) Encoding of Linear codes: Encoding using generator matri- ces, Systematic Encoding (5) Syndrome decoding of linear codes: Cosets, Nearest neighbour decoding, syndrome decoding (6) Bounds for linear codes: Singleton bound and MDS codes, Sphere covering bound, Gilbert-Varshamov bound (7) Important Linear codes: Hamming codes -Encoding and decoding, Binary Golay Codes, Reed-Muller Codes. (8) Cyclic codes: Defini- tion of cyclic codes, Polynomial representation, Generator polynomials and generator matrices, Reciprocal polynomials and parity-check matrices, practical implementation of cyclic codes, de- coding of cyclic codes (9) BCH codes: Definition, Parameters, Decoding BCH codes (10) Reed- Solomon codes: Definition, Parameters, Encoding, Decoding Reed-Solomon codes (10) Low- Density Parity-Check Codes: Definition, Tanner Graph Representation, Constructions, Gallager Decoding Algorithm.

Topics List

Week	Topic	Sections from Moh's book	Prerequisite
1-2	Vector space codes	1	MAT1313, CS2233/2231, or instructor consent.
3-4	Introduction to ring theory	2
5-6	Ring codes	3
7-8	Introduction to algebraic geometry	4
9-10	Algebraic curve Goppa codes	5
11-13	Decoding the geometric Goppa codes	6