## Lecture Notes

Lecture Lecture NotesTopics
1Error Detection Schemes
(.pdf file)
PostNET, ISBNs, UPC, BookLand, Bank Check, Check Sum Schemes
2Introductory Concepts
(.pdf file)
History, Bell Labs, Hamming quote, Repeat Code, Block codes
error detecting, error correcting, single error correcting example
3Combinatorics in Space
(.pdf file)
Mariner 9 Telemetry Specifications
4 Basic Theory
(.pdf file)
Hamming distance, settings, minimum distance, weight
5Math Meets the Bookies
(.pdf file)
6 Linear Codes
(.pdf file)
Linear codes, equivalence of linear codes, generator matrix
parity check matrix, Singleton bound, Hamming codes
7 Decoding Linear Codes
(.pdf file)
Syndromes,Hamming decoding, code cosets, syndrome decoding
8 Finite Fields
(
.pdf file)
Definitions, basic properties, constructions, subfields, automorphisms
9A 2-error correcting code
(.pdf file)
Example of a length 15 binary BCH code
10 Cyclic Codes I
(.pdf file)
Definition, Examples, Rings, Polynomial Rings, Ideals, Principal Rings
Ideals and Cyclic Codes, Generator polynomials
11 Cyclic Codes II
(.pdf file)
Generator matrix, Dual Code, Parity Check Matrix, Syndromes, Error Trapping
12Polynomials
(.pdf file)
Minimal polynomials, Cyclotomic Cosets, factoring xn - 1
13Burst Error Correction
(.pdf file)
Burst Error Correction
14Idempotents & Binary Cyclic Codes
(.pdf file)
Idempotent generators, Minimal Ideals, Examples
15MDS Codes
(.pdf file)
Maximum Distance Separable Codes, Projective Geometry, Ovals, Normal Rational Curves
16Reed-Muller Codes
(.pdf file)
Recursive Definition, (u,u+v)-construction, Properties, Examples, Reed Decoding
Geometric View, Flats in AG(n,2), Mariner 9 Application
17Designs and Codes
(.pdf file)
BIBD's, Examples, Parameters, t-Designs, Derived Designs, Complementary Designs
Designs in Codes, Perfect Codes, Self-Orthogonal Codes, Doubly Even Codes
Self-Dual Codes, Golay Codes, Assmus-Mattson Theorem, Extremal Codes
18Weight Distribution
(.pdf file)
Weight Enumerators, MacWilliams Identity, Codes in Designs, Projective Plane of order 10
Stirling Numbers of the 2nd Kind, Pless Power Moments, Gleason Polynomials
Proof of Assus-Mattson

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