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

A mix of human-written and AI-generated textHuman understanding: all partscs.IT — Information Theorymath.CO — Combinatoricsmath.GR — Group Theory

Contributed by Tejasvi Singh Tomar ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Whether a binary doubly-even self-dual [72,36,16][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 C2C_2, C3C_3, C2×C2C_2\times C_2 or C5C_5. We prove that such a code has no automorphism of order 22, 33 or 55, and hence no nontrivial automorphism. Each such automorphism reduces the problem to a finite computer search. For order 22 the search runs over symmetric bilinear forms on the 4141 self-dual [36,18,8][36,18,8] codes. An automorphism of order 33 leads to a search over labellings of the Hermitian self-dual [24,12,8][24,12,8] codes over F4\mathbb{F}_4 by the extended Golay code. Order 55 reduces to 7×77\times7 unitary matrices over F16\mathbb{F}_{16}. Along the way we classify the Hermitian self-dual [24,12,8][24,12,8] codes over F4\mathbb{F}_4: there are 26 924 93126\,924\,931 of them up to monomial equivalence.

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

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Version history

  1. v1 · 22 pagesSubmitted by Tejasvi Singh TomarPublished on Zenodo: https://doi.org/10.5281/zenodo.23789973CurrentNov 12, 2026