Tel-Aviv University
School of Mathematical Sciences
Department of Applied Mathematics

Applied Mathematics Seminar


Tuesday, March 22nd, 2011

Schreiber 309, 15:10



Shlomo Engelberg

Jerusalem College of Technology

An Introduction to Finite Fields and Non-Linear Coding

Abstract:

We consider linear and non-linear codes. We start by developing a conservation law for codes. We then explain why linear codes, which are easy to understand and implement, are useful when one is interested in protecting data from rarely occurring random errors. By a simple argument, we demonstrate that linear codes are not a good way to protect data from an attacker.

Having ruled out linear codes for this purpose, we take up non-linear codes. We explain what a finite field is and how data can be represented by elements of a finite field. We then consider codes that are non-linear functions of the data -- of the elements of the finite field. We show that quadratic codes suffer from the same drawbacks as linear codes. Next we consider cubic codes. First we show that if all that one is concerned with are attackers, cubic codes are optimal. Then we demonstrate that certain cubic codes provide optimal protection against attackers and some protection against certain relatively common random errors. Many of the results presented in this introduction are due to M. Karpovsky and his co-workers.

This is joint work with Dr. Osnat Keren of Bar Ilan University