• Linear algebra (vector spaces, linear maps, kernel, image, matrices). For example, chapters I--IV of "Linear Algebra" by S. Lang, 3rd ed., Springer Undergraduate Texts in Mathematics.
  • Basic algebra (polynomials with coefficients in a field, basics of finite fields). For example, sections A.3--A.5 of (in the Appendix).
  • Elementary combinatorics (double counting, binomial coefficients, bijective proofs).
  • Elementary probability (expectation and variance of discrete random variables).

Aim of the course
In the context of digital communication, error-correcting codes are mathematical objects that correct errors in noisy and lossy channels. They find several applications in satellite communications, digital audio systems, wireless communications, error correction chips, deep space exploration probes, flash memories, networking, data storage, and in many other contexts.

This course offers an introduction to the mathematical theory of error correction (encoding and decoding information using mathematics). The main aims are:

  • to become familiar with the mathematical methods used to protect digital information from noise, damage, loss, and jamming attacks;
  • to become familiar with the mathematical structure of error-correcting objects;
  • to understand the connections between coding theory and other branches of mathematics (algebra, combinatorics, probability).

Tentative list of topics: communication channels, the Hamming distance, error-correcting codes, bounds, Reed-Solomon codes, Reed-Muller codes, decoding, duality theory of codes, locality, DNA data storage, rank-metric codes.

Alberto Ravagnani (TU/e)