Summary: The original McEliece cryptosystem uses length-$n$ codes over {\bf F}$_2$ with dimension $\geq n-mt$ efficiently correcting $t$ errors where $2^m \geq n$. This paper presents a generalized cryptosystem that uses length-$n$ codes over small finite fields {\bf F}$_{q }$ with dimension $\geq n-m(q-1)t$ efficiently correcting $\lfloor{qt/2}\rfloor$ errors where $q^m \geq n$. Previously proposed cryptosystems with the same length and dimension corrected only $\lfloor{(q-1)t/2}\rfloor$ errors for $q \geq 3$. This paper also presents list-decoding algorithms that efficiently correct even more errors for the same codes over {\bf F}$_{q }$. Finally, this paper shows that the increase from $\lfloor{(q-1)t/2}\rfloor$ errors to more than $\lfloor{qt/2}\rfloor$ errors allows considerably smaller keys to achieve the same security level against all known attacks.