Welcome on the Home Page of the module *Kodierungstheorie*.

Lectures:

Prof. Dr. Hatice Boylan (June, July)

Prof. Dr. Nils-Peter Skoruppa (April, May)

Exercise sessions:

Ali Ajouz M.Sc., Prof. Dr. Hatice Boylan, Prof. Dr. Nils-Peter Skoruppa,

### Classification and preliminaries

This is the second part of the module *Kryptographie und
Kodierungstheorie* whose first part (on cryptography) was given
by Dr. F. Cléry in the last winter term. Accordingly, those who
attended the winter class should continue with this part to obtain
their full credits for the whole course.

However, the second part is independent from the first one and might therefore be of interest for students solely interested in coding theory. This would count then as module \(2+1\) (\(2\) hours lecture and \(1\) hour exercise class per week). The credits given for the \(2+1\) version will depend on your direction of studies; please contact us at the beginning of the course for discussing this question.

The course assumes merely more than linear algebra. A little knowledge
in algebra or number theory can be at times quite helpful, but is not
strictly necessary. The notion of a *finite field* will be
(re)explained in the course.

The course language will be English.

### Contents

From the purely mathematical point of view *Coding Theory* is interested
in subspaces \(C\) (or, more generally, subsets) of the standard vector
space \(F^n\) of \(n\)-vectors with entries from a finite field \(F\)
which optimize two mutually contradictory goals. Namely, the vectors
in \(C\) should be *very different* from each other, but at the
same time \(C\) should have a small co-dimension. That two vectors
are *very different* means that the number of positions where
they differ should be as high as possible. The efficiency of a \(C\)
with respect to these constraints is measured by the *rate*
\(R=R(C)=\frac{\dim_F C}n\) and the *minimal distance*
\(d=d(C)\) which is the minimum of the number of places at which two
different vectors in \(C\) differ.

Such vector spaces, henceforth called *codes*, have quite a
practical impact: they serve for securing information during
transmissions at low cost of energy and speed. Technical examples for
these kind of problems are radio transmissions at long distance (Mars
robots to earth) or audio/video discs which are still readable fast
and reliably though they might carry signs of scratches.

Accordingly, there are two main problems to solve:

- Understanding the set of pairs \((R(C),d(c))\) in the real \(R-d\)-plane, in particular, its borders.
- Finding constructions for efficient codes \(C\).

The course provides an introduction in this kind of problems and the state of art of the basic methods to solve these problems.

Of highly practical interest are codes over the field \(\mathbb{F}_2\) of two elements. We shall therefore often confine ourselves to this case in the course. On the other hand, there are also recent investigations where the field \(F\) is replaced by more general finite rings. If time allows, we shall also follow this thread.

### Literature

J.H. van Lint: Introduction to Coding Theory, GTM 86, Springer 1998

More literature will follow later.