math 5251 - error correcting codes, finite fields, algebraic curves

this is a course about the mathematics of transmiting information. it is not a course about encoding/decoding messages for secrecy (cryptology).

we will primarily use Paul Garrett’s book on the subject. here is an unofficial errata for the book. the book is legally available for free on Garrett’s website. here are some more resources from Garrett. we will also follow Vic Reiner’s notes on the subject, which can be found on his website.

here is information about coding theory using SageMath, a free open-source mathematics software system.

course logistics

grades will be earned through homework assignments, midterms, and a final.

• homework is 50% of grade
• each of two midterms are 15% of grade
• final exam is 20% of grade

homework assignments

there will be 6 homework assignments

homework #	problems	due date
1	PG 1.28, 1.31, 1.33, 2.03, 3.02, 3.05 additional problems	Feb. 1
2	PG 2.04, 3.05, 4.01, 4.02, 4.04, 4.05, 4.06, 4.11 clarification on 4.05	Feb. 15
3	PG 5.01-5.05, 5.08, 6.01, 6.03, 6.07, 6.22, 6.37, 6.49, 6.50, 6.57, 6.52, 6.80, 6.81, 8.17	Mar. 15 Mar. 17
4	PG 6.30, 6.31, 9.11, 9.12, 10.04, 10.08, 10.11, 11.11, 12.06, 12.10, 12.12, 12.14, 12.15	Mar. 29
5	PG 12.01, 12.02, 12.04, 12.14, 12.15, 12.17, 12.19, 12.20, 13.02, 13.05, 13.07, 13.09, 13.10	Apr. 19
6	PG 11.04, 11.05, 14.02, 14.05, 15.03, 15.13	Apr. 26 (one week)

exams

exam #	pdf	due date
1	midterm 1	Feb. 22
2	midterm 2	Apr. 5
final	final exam	May 3

additional resources

some of the following resources may only be available with UMN library access.

• first day of class parlor trick from Richard Ehrenborg,
• two tricks from Todd Mateer
  • using the Hamming code,
  • using Reed-Solomon codes
• a workshop at UMN on using $\LaTeX$ to be held on Feb 10.
• a Quanta article about Reed-Solomon codes
• a link to Shannon’s original paper introducing entropy.
• a 3blue1brown video about Hamming codes

lecture content

this reflects what was actually covered on the given date.

date	rough content
1/18	Intro + PG 3.1
1/23	PG 3.1 + 3.2
1/25	PG 3.2 + 1.4 + 1.5 + 2.2
1/30	PG 3.3
2/1	PG 3.4
2/6	PG 3.4 (ctd) Morse code activity ChatGPT activity
2/8	PG 4.1
2/13	PG 4.2 + 4.4
2/15	PG 4.4 (ctd) Max likelihood activity
2/20	PG 2.2 + 4.4 + 4.5
2/22	Snow day
2/27	PG 5.1 + 5.3 Polynomial division review
3/1	PG 6.5
3/6	See Canvas for video, please watch before 3/15
3/13	PG 6.5 ctd
3/15	PG 6.9, 6.10, start chapter 10
3/20 - 3/21	Continue chapter 10 following Reiner’s notes
3/27	Linear codes and linear algebra, start PG 12
3/29	Sage exercise. (create CoCalc account before class)
4/3	Linear codes and linear algebra (PG 12/Appendix A.1), Gaussian binomial
4/5	Linear codes and linear algebra ctd (PG 12/Appendix A.2)
4/10	Encoding with linear codes and dual codes (PG 12.7)
4/12	Decoding with linear codes and dual codes (PG 12.8)
4/17	Bounds on codes and cyclic codes (PG 13, 14.2)
4/19	Cyclic codes ctd
4/24	Finite fields and primitive roots (PG 11, 15)
4/26	Finite fields and primitive roots ctd
5/1	Reed-Solomon codes (PG 17, last day of instruction)

questions

my students ask great questions. here are the answers to some of them.