Some aspects of coding theory between probability, algebra, combinatorics and complexity theory

Abstract

The aim of this survey report is to draw attention to some recent developments which seem to have changed the face of Coding Theory completely. While this area of applicable algebra -which has strongly been influenced [3] by hard problems of communications engineering- during the last two decades has become a main part of Combinatorics, reachign from Finite Geometries to Representation Theory [18], it has never been fully accepted as a part of Algebra itself — the reasons for this being manifold. On the one hand, Coding Theory can easily be mistaken for a part of Linear Algebra, while on the other hand a non-typical feature distinguishes it from the main concept of modern and classical algebra: The properties of codes are "basis-dependent" so that the many tools of "basis-free" algebra are not always helpful.

Due to some very recent publications ([13],[19],[29]) this situation may be changing very soon, as the interaction between these fields has provided new insights into both:

Results from Algebraic Geometry permit the construction of codes, which are better than those known before, while very well-known bounds on codes in turn improve Weil's bound for the number of points on a curve over a finite field.

On the other hand, a very recent paper [9] shows that the construction of extremely good codes is possible by rather elementary means.

The aim of this survey report is to introduce a general mathematical audience to the background, eventually leading to these new developments. As is usual for a survey the author has included results from many different fields, not just from his own one. Thus, this report hopefully is in the spirit of the classical understanding of research — providing a collection of material which is not even contained in the most recent book [30] on Coding Theory.