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Tel Aviv University
School of Mathematical Sciences
Applied Mathematics Seminar

Date:          Tuesday March 22, 2011, 15:10
Place:             Schreiber Bldg, Room 309

Speaker:     Shlomo Engelberg
Jerusalem College of Technology

Title: 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

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