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Algebraic Coding Theory
MATH 4012 Spring 2026
Dr. Hunter Lehmann
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\(\require{mathtools} \setcounter{MaxMatrixCols}{15} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\cC}{\mathcal{C}} \newcommand{\F}{\mathbb{F}} \newcommand{\GF}{\mathrm{GF}} \DeclareMathOperator{\Span}{Span} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\rk}{rk} \DeclareMathOperator{\wt}{wt} \DeclareMathOperator{\im}{im} \makeatletter \@ifundefined{char}{ \DeclareMathOperator{\char}{char} }{} \makeatother \DeclareMathOperator{\GRS}{GRS} \DeclareMathOperator{\RS}{RS} \DeclareMathOperator{\BCH}{BCH} \newcommand{\transpose}[1]{#1^{\top}} \newcommand{\by}{\mathbf{y}} \newcommand{\bc}{\mathbf{c}} \newcommand{\bx}{\mathbf{x}} \newcommand{\bm}{\mathbf{m}} \newcommand{\bs}{\mathbf{s}} \newcommand{\be}{\mathbf{e}} \newcommand{\bu}{\mathbf{u}} \newcommand{\bv}{\mathbf{v}} \newcommand{\bh}{\mathbf{h}} \newcommand{\br}{\mathbf{r}} \newcommand{\bzero}{\mathbf{0}} \newcommand{\balpha}{\boldsymbol{\alpha}} \renewcommand{\vec}[1]{\boldsymbol{#1}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \newcommand{\fillinmath}[1]{\mathchoice{\underline{\displaystyle \phantom{\ \,#1\ \,}}}{\underline{\textstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptscriptstyle\phantom{\ \,#1\ \,}}}} \)
Front Matter
Course Documents
Syllabus
Course Information
My Philosophy
Course Structure
Course Grading
Course Websites
Other Course Policies
Grading Specifications
General Expectations for All Work
Significant Errors and Minor Errors
Specifications for Assignments
Engagement Credit Items
Specifications for Exams
Specifications for the Video Project
Course Calendar
Video Project
Video Project Requirements
Team Contract
Report
Peer Review
Deadlines
Course Resources
Calculation Tools
Matrices over Finite Fields
Bounds
Lagrange Interpolation for Reed Solomon
Polynomial Operations over Finite Fields
Sugiyama Euclidean Algorithm
Welch Berlekamp System
Definitions and Algorithms
Definitions
Algorithms
Results
Results
Course Notes
Week 1
Tuesday 1/13
Thursday 1/15
Week 2
Tuesday 1/20
Thursday 1/22
Week 3
Tuesday 1/27
Thursday 1/29
Week 4
Tuesday 2/3
Thursday 2/5
Week 5
Tuesday 2/10
Thursday 2/12
Week 6
Tuesday 2/17
Thursday 2/19
Week 7
Tuesday 2/24
Thursday 2/26
Week 8
Tuesday 3/3
Thursday 3/5
Week 9
Tuesday 3/10
Thursday 3/12
Week 10
Tuesday 3/17
Thursday 3/19
Week 11
Tuesday 3/24
Thursday 3/26
Week 12
Tuesday 3/31
Thursday 4/2
Daily Prep Activities
Daily Prep 2
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Exercises
Daily Prep 3
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Exercises
Daily Prep 4
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Exercises
Daily Prep 5
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Daily Prep 6
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Daily Prep 7
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 8
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 9
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 10
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 11
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 12
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 13
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 14
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 15
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 16
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 17
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 18
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Daily Prep 19
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 20
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
Daily Prep 21
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms and Results
Daily Prep 22
Learning Objectives
Basic Learning Objectives
Advanced Learning Objectives
Resources for Learning
Important Terms
In-Class Activities
Introduction to Codes
Distance of a Code and Error Correction
The Hamming Code
Linear Algebra with Finite Fields
Linear Codes
1-Error Correcting Codes
Decoding Linear Codes
Bounds on Codes
The Gilbert-Varshamov Bound
Irreducible Polynomials & Building Finite Fields
Order, Characteristic, and Zech’s Logs
Frobenius Maps & a 2-Error Correcting Code
Quadratics over GF(16) and a First Look at Reed-Solomon Codes
GRS and RS Codes
GRS and RS Codes Take 2
Concatenated Codes and BCH Codes
Intro to GRS Decoding
Welch-Berlekamp Decoding
Properties of Minimal Polynomials & Finite Field Isomorphisms
Reviewing Codes Derived from RS Codes & More Minimal Polynomials
Cyclic Codes
Error Trapping for Cyclic Codes
Handouts
GRS & RS Codes
GRS Decoding Schemes
Weekly Practice
Weekly Practice 1
Weekly Practice 2
Weekly Practice 3
Weekly Practice 4
Weekly Practice 5
Weekly Practice 6
Weekly Practice 7
Weekly Practice 8
Weekly Practice 9
Weekly Practice 10
Weekly Practice 11
Weekly Practice 12
Weekly Practice 13
Problem Sets
Problem Set M1
Problem Set M2
Problem Set M3
Problem Set M4
Problem Set M5A
Problem Set M5B
Problem Set M6
Algebraic Coding Theory
MATH 4012 Spring 2026
Dr. Hunter Lehmann
School of Mathematics
Georgia Institute of Technology
Last Updated: April 21, 2026
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