The automorphism group of the doubly-even [72,36,16] code can only be of order 1, 3 or 5

TL;DR

This paper proves that the automorphism group of the extremal doubly-even binary code of length 72 and minimum distance 16 can only have order 1, 3, or 5, by ruling out larger automorphism groups with involutions.

Contribution

It demonstrates that such a code cannot have an automorphism group containing involutions or fixed point-free involutions, narrowing possible automorphism group orders to 1, 3, or 5.

Findings

Automorphism group cannot contain fixed point-free involutions.

Automorphism group cannot contain involutions with fixed points.

Possible automorphism group orders are 1, 3, or 5.

Abstract

We prove that a putative $[72,36,16]$ code is not the image of linear code over $\ZZ_4$, $\FF_2 + u \FF_2$ or $\FF_2+v\FF_2$, thus proving that the extremal doubly even $[72,36,16]$-binary code cannot have an automorphism group containing a fixed point-free involution. Combining this with the previously proved result by Bouyuklieva that such a code cannot have an automorphism group containing an involution with fixed points, we conclude that the automorphism group of the $[72,36,16]$-code cannot be of even order, leaving 3 and 5 as the only possibilities.