  Guralnick, Kantor, Kassabov and Lubotzky (J. Eur. Math. Soc. 13.2, 2011,
391-458) [GKKL] give 3-generator 7-relator presentations of $A_n$ and $S_n$
with bit-length $O(\log n)$ for $n\geq5$. This is the best possible bit-length,
since $\Omega(\log n)$ bits are required to specify the integer $n$ in the
input. However, the generators do not satisfy the relations.
  This paper considers the relevant arguments given in [GKKL], identifies where
the errors occur, and shows how they can be fixed in order to recover this
result. The presentations are available in Magma.

Group Theory