\chapter{Conclusions and future work}
Error-correcting codes and cryptography play important roles in information communication.
Generalized Reed-Solomon codes and cryptography systems based on discrete logarithms are 
representatives of these areas, respectively.
In this dissertation, we have studied the deep hole problem of generalized Reed-Solomon codes 
and the discrete logarithm problem over finite fields.

In the first part,  we classify deep holes for Generalized Reed-Solomon codes $RS_q(D,k)$. 
Specifically, we give a characterization of deep holes when
$q$ is an odd prime, $|D| > k \geqslant \frac{p-1}{2}$. Generalizing the result
to finite fields of composite order is a possible future work.

In the second part, we study the validation of the heuristics made 
in the quasi-polynomial time algorithm solving the discrete logarithms
in the small characteristic fields \cite{BGJT13v1}.  We find that
the heuristics are problematic in the cases of non-Kummer extensions. 
We propose a few modifications to the algorithm, including
some extra requirements for the polynomials $ h_0 $ and $ h_1 $,
and a trap-avoiding descent strategy.  The modified algorithm 
relies on three improved heuristics. 
\begin{proposition}
  If  Heuristics \ref{HSmith}, \ref{Hh0h1} and \ref{Hdescent} hold, then 
the discrete logarithm problem 
over $ \F_{q^k} $ ($ k<q $)  can be solved in time $ q^{O(\log (k))}  $.
%In general,  the discrete logarithm problem 
%over $ \F_{q^k} $   can be solved in time $ exp({poly(\log (q^k))}  $.
\end{proposition}
We believe that proving 
(or disproving) these heuristics offers 
interesting open problems that will help to understand the effectiveness
of the new algorithm. Besides, it is interesting to explore the applications of the
idea used to attack the discrete logarithms over finite fields to other problems. 