The first important result is in the paper: Pless, Vera; Thompson, John G.
$17$ does not divide the order of the group of a $(72,\,36,\,16)$ doubly
even code.
IEEE Trans. Inform. Theory 28 (1982), no. 3, 537--541.
Authors' summary: "It is an interesting open question whether an
extremal (72,36,16) doubly even code $C$ exists.
In an article by J. H. Conway
and the first author [Discrete Math. 38 (1982), no. 2--3, 143--156] the
odd prime numbers which can divide the order of the group of $C$ were
determined. The largest of these, 23, was eliminated by finding weight
12 vectors in 384 codes [the first author, IEEE Trans. Inform. Theory 28
(1982), no. 1, 113--117; MR 83g:94026].
The next largest prime remaining is 17. It is shown that 17 is also not possible by reducing the problem to
the consideration of $16*17\sp 3$ codes and then finding a weight 12
vector, by computer, in each of these codes."
The second result is in the paper:
Huffman, W. Cary(1-LYLCH); Yorgov, V. Y.(BG-HPI)
A $[72,36,16]$ doubly even code does not have an automorphism of orde=
r $11$.
IEEE Trans. Inform. Theory 33 (1987), no. 5, 749--752.
A code with the parameters given in the title, if such a code exists,
would be a third member in an interesting sequence of possible codes
with
parameters $[24m,12m,4m+4]$ which begins with the Golay code $(m=1)$
and the quadratic residue code $(m=2)$ [N. J. A. Sloane, same
journal 19 (1973), no. 2, 251; MR 54 #9843]. A consequence of the
present paper is that the order of the automorphism group of such a code
can
have no prime divisors other than 2, 3, 5, and 7. An automorphism of
order 11 must have 6 orbits of length 11 on the coordinate positions
and must
fix the remaining 6 coordinates. Using this information, it is possible
to derive from a code $C$ as in the title two smaller codes, a self-dual
$[12,6,4]$ binary code and a $[6,3,4]$ code over $\roman{GF}(1024)$
which is self-dual with respect to a Hermitian form. The authors describe
these two codes and then study the possible ways of reconstructing $C$
from them. A computer calculation gives the final contradiction.
In addition Stefka Bouyuklieva has shown that any automorphism of order
2 can't have any fixed point (preprint: On the automorphisms of order 2
with fixed points for the extremal self-dual codes of length 24m).