cs.IT — Information Theory

The automorphism group of an extremal [72,36,16] code is trivial

Contributed by Tejasvi Singh Tomar

Whether a binary doubly-even self-dual $[72,36,16]$ code exists is a question raised by Sloane in 1973. A series of papers, surveyed by Borello, shows that the automorphism group of such a code is trivial or isomorphic to $C_2$, $C_3$, $C_2\times C_2$ or $C_5$. We prove that such a code has no automorphism of order $2$, $3$ or $5$, and hence no nontrivial automorphism. Each such automorphism reduces the problem to a finite computer search. For order $2$ the search runs over symmetric bilinear forms on the $41$ self-dual $[36,18,8]$ codes. An automorphism of order $3$ leads to a search over labellings of the Hermitian self-dual $[24,12,8]$ codes over $\mathbb{F}_4$ by the extended Golay code. Order $5$ reduces to $7\times7$ unitary matrices over $\mathbb{F}_{16}$. Along the way we classify the Hermitian self-dual $[24,12,8]$ codes over $\mathbb{F}_4$: there are $26\,924\,931$ of them up to monomial equivalence.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23789973