Finite Fields and Coding Theory

General Information

• Topics: This course is an introduction to finite fields, emphasizing their structure and applications to coding theory. The influence of computational problems will be considered. Therefore, this course centers around three main issues:
  • the mathematics of finite fields,
  • the applications to coding theory, and
  • the associated computational problems.
• Tentative lecture schedule (html file) as of December 2017 (before classes start).
• Textbook: There is no official textbook for this course. However, for most of the lectures, we will have notes available. They were prepared by students of a previous version of the course and checked by the professor. We will extensively use material from the following books:
  • ``Introduction to Finite Fields and Their Applications'', by R. Lidl and H. Niederreiter, Cambridge University Press, 1994.
  • ``Lectures on Finite Fields and Galois Rings'', by Zhe-Xian Wan, World Scientific Publishing Company, 2003.
  Other books that we may use are:
  • ``Algebraic Coding Theory'', by E. Berlekamp, Aegean Park Press, 1984.
  • ``Modern Computer Algebra'', by J. von zur Gathen and J. Gerhard, 3rd edition, Cambridge University Press, 2013.
  • ``Coding Theory and Cryptography: the Essentials'', by D.R. Hankerson, D.G Hoffman, D.A. Leonard, C.C. Lindner, K.T. Phelps, C.A. Rodger and J.R. Wall, Marcel Dekker Inc., 2000.
  • ``Finite Fields'', by R. Lidl and H. Niederreiter, Cambridge University Press, 1997.
  • ``The Theory of Error-Correcting Codes'', F.J. MacWilliams and N.J.A. Sloane, North-Holland, Elsevier Science, 1977.
  • ``Finite Fields for Computer Scientists and Engineers'', R. McEliece, Kluwer Academic Press, 1987.
• Prerequisites: MATH 2100 or 3101 or 2108 or equivalent; or permission of the School.
• Classes begin: Thursday, January 5, 2017.
  Classes end: Thursday, April 6, 2017.
  
• Term mark: There will be three assignments:

  Assignment	Hand-out Date	Due Date	Contents	Worth
  1	January 24	February 14	Linear, Hamming and BCH Codes	12%
  2	February 14	March 7	Structure of Finite Fields	12%
  3	March 7	March 28	Polynomials; Cyclic Codes	12%

  You are expected to do all the assignments. No make up, early, or delayed assignments. Any missing assignments will be counted as zero.
  The assignments must be your own work. In particular, you must cite everything you are taking from the literature or that you discussed with someone else.
  Total portion of the final mark in assignments: 36%. There will be a midterm test in class on Thursday March 16; worth of midterm test: 14%.

• Evaluation:
  • either: term mark is 50%; final exam is 50%,
  • or: term mark is 50%; final exam is 30%; project and presentation 20%.
  The due date for the project is Tuesday April 4, 2017. You must have a meeting with the instructor to discuss your project before Thursday March 2, 2017. The presentations will take place in early April; the day will be arranged later, possibly on Thursday April 6 at class time. Graduate students must take project and presentation option; undergraduate students may choose to have the project and presentation option. This option involves writing a brief report (10-15 pages) plus a presentation (25-30 minutes). Here is a list of possible projects (html file). This is not a closed list of possibilities. Indeed, students interested in topics for the project and presentation that are not in this list should consult the professor about the feasibility of such a topic. Essentially, any topic related with finite fields and/or coding theory of interest to the student could be considered for the project after approval by the professor.
  
• Withdrawal: The last day for withdrawal from the course without academic penalties is the last day of classes.
• Academic Accommodation:
You may need special arrangements to meet your academic obligations during the term. For an accommodation request the processes are as follows:
• Pregnancy and Student Parental Leave: write to me with any requests for academic accommodation during the first two weeks of class, or as soon as possible after the need for accommodation is known to exist. For more details see the Parental Leave Guide.
• Academic Accommodations for Students with Disabilities: The Paul Menton Centre for Students with Disabilities (PMC) provides services to students with Learning Disabilities (LD), psychiatric/mental health disabilities, Attention Deficit Hyperactivity Disorder (ADHD), Autism Spectrum Disorders (ASD), chronic medical conditions, and impairments in mobility, hearing, and vision. If you have a disability requiring academic accommodations in this course, please contact PMC at 613-520-6608 or pmc@carleton.ca for a formal evaluation. If you are already registered with the PMC, contact your PMC coordinator to send me your Letter of Accommodation at the beginning of the term, and no later than two weeks before the first in-class scheduled test or exam requiring accommodation (if applicable). After requesting accommodation from PMC, meet with me to ensure accommodation arrangements are made. Please consult the PMC website for the deadline to request accommodations for the formally-scheduled exam (if applicable).
• Religious obligation: write to me with any requests for academic accommodation during the first two weeks of class, or as soon as possible after the need for accommodation is known to exist. For more details see the Religious Observation Guide.

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Contact Information:

Office Hours are Thursdays 1:05-1:55 pm in HP 4372.
e-mail: daniel@math.carleton.ca