\titleofpresentation{Asymptotically good codes with transitive automorphism groups}

\presenter{Henning Stichtenoth}  %  Name just one speaker, please; collaborators
			%  can be credited in the body of the abstract.

%  [Body of abstract goes here.]
One of the main open problems in algebraic coding theory is the question whether 
the class of cyclic codes is asymptotically good. This means: Is there a sequence
of cyclic codes $(C_i)_{i\ge 0}$ over the finite field $\mathbb{F}_q$ whose parameters
$[n_i,k_i,d_i]$ (length, dimension, minimum distance) satisfy

$$n_i \rightarrow \infty, \ \ \liminf_{i \rightarrow \infty} (k_i/n_i) > 0 \ \ {\rm and} \ \
\liminf_{i \rightarrow \infty} (d_i/n_i) > 0.$$

Cyclic codes $C \subseteq \mathbb{F}_q^n$ have the property that their  
automorphism group is a transitive subgroup of the symmetric group $S_n$. Hence
it is natural to consider the class of {\it transitive} codes. Our main result is:
 
\begin{theorem*} The class of codes over $\mathbb{F}_q$ (where $q$ is a square) 
with transitive automorphism group is asymptotically good.
\end{theorem*} 

The proof of the theorem uses Goppa's algebraic-geometric (AG) codes over function fields.
The main ingredient is a new asymptotically good tower of function fields 
$F_0 \subseteq F_1 \subseteq F_2 \subseteq \ldots$  over $\mathbb{F}_q$ with the property that
all extensions $\mathbb{F}_n/ \mathbb{F}_0$ are galois. With a careful choice of the 
divisors that define the corresponding AG codes one then obtains a transitive action 
of the Galois group on the codes. 
%\begin{ourbiblio}{99}  %  Delete this environment if you
		       %  aren't giving a bibliography.

%\bibitem{yourname1} One paper, keyed to your name to avoid conflicts 
%with other abstracts;

%\bibitem{yourname2} Another paper;

%\bibitem{yourname3} Etc.

%\end{ourbiblio}

\presenteraddress{Sabanc{\i} University}  % The name of a university is fine.

\email{henning@sabanciuniv.edu}

\web{www.stat-math.uni-essen.de/~stichtenoth/henning/}  % Those without homepages should delete this.

