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,

Eintrag im LSF

### 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