TejasviTomar
October 2026 · Preprint

The automorphism group of an extremal [72,36,16] code has order at most 2

Tejasvi Singh Tomar

In plain words

In 1973 Neil Sloane asked whether a certain error-correcting code exists: 72 bits long, with any two codewords differing in at least 16 places, the most its kind allows. Its 24-bit relative, the Golay code, protected the data Voyager sent from Jupiter and Saturn. The question is still open. From 1982 on, John Conway, Vera Pless, John Thompson and others narrowed the symmetries such a code could have to five cases. Exhaustive computer searches rule out three: the code, if it exists, has no symmetry or just one, which swaps its positions in pairs.

Abstract

Whether a binary doubly-even self-dual [72,36,16] code exists is a question raised by Sloane in 1973. The automorphism group of such a code is known to be trivial or isomorphic to C2, C3, C2 × C2 or C5. We exclude C5, C2 × C2 and C3. For an automorphism of order 5 the fixed subcode is e8 ⊕ e8, and minimum distance 16 becomes a condition on a 7 × 7 unitary matrix over F16. For a subgroup ⟨a, b⟩ ≅ C2 × C2 the code is determined by a symmetric bilinear form on the contracted fixed code of a, which is one of the 41 self-dual [36,18,8] codes. After a normalisation the form ranges over an affine space of dimension 63, 64 or 65, and minimum distance 16 becomes one syndrome condition for each word of weight 8, 10 or 12. For an automorphism of order 3 the code is determined by a Hermitian self-dual [24,12,8] code over F4 and a labelling of its coordinates by the points of the extended Golay code. We classify these codes over F4: there are 26 924 931 of them up to monomial equivalence. Exhaustive computer searches show that none of the three reduced problems has a solution. Each search is confirmed by a second implementation, and tests on known codes recover them. Consequently the automorphism group of an extremal [72,36,16] code is trivial or of order 2.

Two heptagons side by side on white, with no labels. Each is made of seven small pentagons of dots, with a thin straight line between every two pentagons and a small square in the middle. Left, in pale pink: the three lines of one triangle of pentagons are bold magenta, and its fifteen dots and the square are filled magenta. Right, in pale teal: a different triangle is bold teal, and its fifteen dots and the square are filled teal.
Illustration of the order-5 case (Theorem A). An automorphism of order 5 of an extremal [72,36,16] code would split its 72 positions into 14 cycles of five (the pentagons, each turned one step by the automorphism) and two fixed points (the squares). The codewords that the automorphism leaves unchanged would form two copies of the code e₈, one on each side, each on seven cycles and one fixed point. The seven cycles of a side are the points of a Fano plane, whose seven lines are drawn as triangles: every two cycles lie on exactly one line. In colour are two of these codewords, one on each side, each made of the three cycles on a line and the fixed point. Each has 16 ones, the fewest a nonzero codeword can have. The rest of the code would be given by a 7×7 unitary matrix over the field with 16 elements, and the paper’s exhaustive search shows that no such matrix keeps every nonzero codeword at 16 ones or more.