Math 567: Introduction to Coding Theory
MWF 10AM-11AM, EH4096




Instructor: Benjamin Linowitz
Office: 1854 East Hall
Email: linowitz@umich.edu
Office hours: Monday, Tuesday and Wednesday 1-2pm and by appointment


Textbook: The official textbook for this course is

-van Lint, J. H. Introduction to coding theory, 3rd ed. Graduate Texts in Mathematics, 86. Springer-Verlag, Berlin, 1999.

Our goal will be to cover the first 6 chapters of this book. If time permits, we will then cover special topics depending on the interest of the students. Although this book is considered a classic, it is somewhat tersely written. Other references which may be of use are:

-Adamek, J. Foundations of Coding Theory, Wiley 1991

-Huffman, W. C. and Pless, V. Fundamentals of error-correcting codes. Cambridge University Press, Cambridge, 2003

-Pretzel, O. Error-correcting codes and finite fields. Student ed. Oxford: Clarendon Press


Prerequisites: A prior course in linear algebra will be essential (i.e. Math 217, 417, 419 or 513). In particular I will assume that all students are comfortable with vector spaces over the real numbers, subspaces, linear transformations and the matrix of a linear transformation. During the first month of the course we will study objects called finite fields, at which point we will begin to study vector spaces over finite fields. While we will review the necessary topics from linear algebra as they arise, it will be difficult for students not having taken a previous course in linear algebra to keep up with this level of abstraction.

A previous course in abstract algebra (covering the basic theory of groups, rings and fields, especially polynomial rings and finite fields) will also be helpful, though I will not assume any prior knowledge and will spend the first month or so of the course developing the relevant theory. These topics arise not only in coding theory but in virtually every area of pure mathematics, hence this will be time well spent.


Exams: There will be two in-class midterms and a final exam. The first midterm will be held on Monday February 11. It will cover the background material from abstract and linear algebra which we will be reviewing in class for the first month or so. The second midterm will be held on Monday March 25.


Homework: There will be around 7 homework assignments this term. In general, each assignment will be due two weeks after it was distributed, at which point the next assignment will be distributed. While we are reviewing background material from abstract algebra during the first month of the term however, we will have weekly homework assignments.

Homework 1 (Due Wednesday January 23)
Homework 2 (Due Wednesday February 13)
Homework 3 (Due Friday March 1)
Homework 4 (Due Wednesday March 20)
Homework 5 (Due Wednesday March 27)
Homework 6 (Due Friday April 5)
Homework 7 (Due Monday April 22)


Grading:

Homework: 50%
Midterm 1: 10%
Midterm 2: 20%
Final Exam: 20%



Related Lecture Notes:

Keith Conrad's excellent notes on Characters on Finite Abelian Groups.


Related Research Papers:

-A generalization of the Hansen-Mullen conjecture on irreducible polynomials over finite fields by Daniel Panario and Georgios Tzanakis.

-Milnor's construction of 16-dimensional flat tori from self-dual lattices.