The automorphism group of a self-dual [72,36,16] code does not contain S_3, A_4, or D_8

A computer calculation with Magma shows that there is no extremal self-dual binary code C of length 72, whose automorphism group contains the symmetric group of degree 3, the alternating group of degree 4 or the dihedral group of order 8. Combining this with the known results in the literature one obtains that Aut(C) has order at most 5 or isomorphic to the elementary abelian group of order 8.


Introduction
Let C = C ⊥ ≤ F n 2 be a binary self-dual code of length n. Then the weight wt(c) := |{i | c i = 1}| of every c ∈ C is even. When in particular wt(C) := {wt(c) | c ∈ C} ⊆ 4Z, the code is called doubly-even. Using invariant theory, one may show [10] that the minimum weight d(C) := min(wt(C \ {0})) of a doubly-even self-dual code is at most 4 + 4 n 24 . Self-dual codes achieving this bound are called extremal. Extremal self-dual codes of length a multiple of 24 are particularly interesting for various reasons: for example they are always doubly-even [12] and all their codewords of a given nontrivial weight support 5-designs [2]. There are unique extremal self-dual codes of length 24 (the extended binary Golay code G 24 ) and 48 (the extended quadratic residue code QR 48 ) and both have a fairly big automorphism group (namely Aut(G 24 ) ∼ = M 24 and Aut(QR 48 ) ∼ = PSL 2 (47)). The existence of an extremal code of length 72 is a long-standing open problem [13]. A series of papers investigates the automorphism group of a putative extremal self-dual code of length 72 excluding most of the subgroups of S 72 . The most recent result is contained in [3] where the first author excluded the existence of automorphisms of order 6. In this paper we prove that neither S 3 nor A 4 nor D 8 is contained in the automorphism group of such a code. The method to exclude S 3 (which is isomorphic to the dihedral group of order 6) is similar to that used for the dihedral group of order 10 in [8] and based on the classification of additive trace-Hermitian self-dual codes in F 12 4 obtained in [7]. For the alternating group A 4 of degree 4 and the dihedral group D 8 of order 8, we use their structure as a semidirect product of an elementary abelian group of order 4 and a group of order 3 and 2 respectively. By [11] we know that the fixed code of any element of order 2 is isomorphic to a self-dual binary code D of length 36 with minimum distance 8. These codes have been classified in [1]; up to equivalence there are 41 such codes D. For all possible liftsD ≤ F 72 2 that respect the given actions we compute the codes E :=D A4 and E :=D D8 respectively. We have respectively only three and four such codes E with minimum distance ≥ 16. Running through all doubly-even A 4 -invariant self-dual overcodes of E we see that no such code is extremal. Since the group D 8 contains a cyclic group of order 4, say C 4 , we use the fact [11] that C is a free F 2 C 4 -module. Checking all doubly-even self-dual overcodes of E which are free F 2 C 4 -modules we see that, also in this case, none is extremal. The present state of research is summarized in the following theorem. 2. The symmetric group of degree 3.

2.1.
Preliminaries. Let C be a binary self-dual code and let g be an automorphism of C of odd prime order p. Define C(g) := {c ∈ C | c g = c} and E(g) the set of all the codewords that have even weight on the cycles of g. From a module theoretical point of view, C is a F 2 g -module and C(g) = C · (1 + g + . . . + g p−1 ) and E(g) = C · (g + . . . + g p−1 ). In [9] Huffman notes (it is a special case of Maschke's theorem) that In particular it is easy to prove that the dimension of E(g) is (p−1)·c 2 where c is the number of cycles of g. In the usual manner we can identify vectors of length p with polynomials in Q : The weight of a polynomial is the number of nonzero coefficients. Let P ⊂ Q be the set of all even weight polynomials. If 1+x+. . .+x p−1 is irreducible in F 2 [x] then P is a field with identity x + x 2 + . . . + x p−1 [9]. There is a natural map that we will describe only in our particular case in the next section, from E(g) to P c . Let us observe here only the fact that, if p = 3, then 1 + x + x 2 is irreducible in F 2 [x] and P is isomorphic to F 4 , the field with four elements. The identification is the following: 0 000 ω 110 1 011 ω 101 2.2. The computations for S 3 . Let C be an extremal self-dual code of length 72 and suppose that G ≤ Aut(C) with G ∼ = S 3 . Let σ denote an element of order 2 and g an element of order 3 in G. By [6] and [5], σ and g have no fixed points. So, in particular, σ has 36 2-cycles and g has 24 3-cycles. Let us suppose, w.l.o.g. that As we have seen in Section 2.1, where E(g) is the subcode of C of all the codewords with an even weight on the cycles of g, of dimension 24. We can consider a map extending the identification P ∼ = F 4 , stated in Section 2.1, to each cycle of g. Again by [9], E(g) ′ := f (E(g)) is an Hermitian self-dual code over F 4 (that is Note that this action is only F 2 -linear. In particular, the subcode fixed by σ, say The code X := π(E(g) ′ (σ)) := {(ǫ 1 , . . . , ǫ 12 ) ∈ F 12 4 | (ǫ 1 , ǫ 1 , . . . , ǫ 12 , ǫ 12 ) ∈ E(g) ′ } is an additive trace-Hermitian self-dual (12, 2 12 , ≥ 4) 4 code such that Proof. For γ, ǫ ∈ X the inner product of their preimages in E(g) ′ (σ) is is obtained from X as stated. All additive trace-Hermitian self-dual codes in F 12 4 are classified in [7]. There are 195, 520 such codes that have minimum distance ≥ 4 up to monomial equivalence. An exhaustive search with Magma (of about 7 minutes CPU on an Intel(R) Xeon(R) CPU X5460 @ 3.16GHz) shows that the minimum distance of φ(X ) is ≤ 6, for each of the 195, 520 additive trace-Hermitian self-dual (12, 2 12 , ≥ 4) 4 codes. But E(g) ′ should have minimum distance ≥ 8, a contradiction. So we proved the following.
Theorem 2. The automorphism group of a self-dual [72, 36, 16] code does not contain a subgroup isomorphic to S 3 .
3. The alternating group of degree 4 and the dihedral group of order 8.
3.1. The action of the Klein four group. For the alternating group A 4 of degree 4 and the dihedral group D 8 of order 8 we use their structure Then by [6] and [5] all non trivial elements in H act without fixed points and we may replace C by some equivalent code so that denote the bijection between the fixed space of g and F 36 2 and Remark 2. The centraliser C S72 (g) ∼ = C 2 ≀ S 36 of g acts on the set of fixed points of g. Using the isomorphism π 1 we obtain a group epimorphism which we again denote by π 1 π 1 : C S72 (g) → S 36 with kernel C 36 2 . Similarly we obtain the epimorphism π 3 : C S36 (π 1 (h)) → S 18 .
The normalizer N S72 ( g, h ) acts on the set of g, h -orbits which defines a homomorphism π 2 : N S72 ( g, h ) → S 18 .
Let us consider the fixed code C(g) which is isomorphic to By [11], the code π 1 (C(g)) is some self-dual code of length 36 and minimum distance 8. These codes have been classified in [1]; up to equivalence (under the action of the full symmetric group S 36 ) there are 41 such codes. Let  and π 2 (σ) ∈ Aut(π 3 (D(π 1 (h)))) .

Remark 5.
With Magma we use the algorithm given in Lemma 1 to compute that there are exactly 25, 299 G 36 -orbits on D, represented by, say, X 1 , . . . , X 25,299 .
Remark 7. For all 1 ≤ i ≤ 25, 299, 1 ≤ j ≤ 64 we compute the code All three groups contain a unique conjugacy class of subgroups conjugate in S 72 to A 4 (which is normal for E 1 and E 3 ).
These computations took about 26 hours CPU, using an Intel(R) Xeon(R) CPU X5460 @ 3.16GHz. Corollary 1. The code C(g) + C(h) + C(gh) is equivalent under the action of G to one of the three codes E 1 , E 2 or E 3 .
Let E be one of these three codes. The group A 4 acts on V := E ⊥ /E with kernel g, h . The space V is hence an F 2 σ -module supporting a σ-invariant form such that C is a self-dual submodule of V. As in Section 2.1 we obtain a canonical decomposition V = V(σ) ⊥ W where V(σ) is the fixed space of σ and σ acts as a primitive third root of unity on W.
For E = E 1 or E = E 2 we compute that V(σ) ∼ = F 4 2 and W ∼ = F 8 4 . For both codes the full preimage of every self-dual submodule of V(σ) is a code of minimum distance < 16.
For E = E 3 the dimension of V(σ) is 2 and there is a unique self-dual submodule of V(σ) so that the full preimage E 3 is doubly-even and of minimum distance ≥ 16. The element σ acts on 10 4 . The code C is a preimage of one of the 58, 963, 707 maximal isotropic F 4 -subspaces of the Hermitian F 4 -space E ⊥ 3 /E 3 . The unitary group GU (10, 2) of E ⊥ 3 /E 3 ∼ = F 10 4 acts transitively on the maximal isotropic subspaces. So a quite convenient way to enumerate all these spaces is to compute an isometry of E ⊥ 3 /E 3 with the standard model used in Magma and then compute the GU (10, 2)-orbit of one maximal isotropic space (e.g. the one spanned by the first 5 basis vectors in the standard model). The problem here is that the orbit becomes too long to be stored in the available memory (4GB). So we first compute all 142, 855 one dimensional isotropic subspaces E 3 /E 3 ≤ F4 E ⊥ 3 /E 3 for which the code E 3 has minimum distance ≥ 16. The automorphism group Aut(E 3 ) = Aut(E 3 ) acts on these codes with 1, 264 orbits. For all these 1, 264 orbit representatives E 3 we compute the 114, 939 maximal isotropic subspaces of E ⊥ 3 /E 3 (as the orbits of one given subspace under the unitary group GU (8,2) in Magma) and check whether the corresponding doubly-even self-dual code has minimum distance 16. No such code is found.
Note that the latter computation can be parallelised easily as all 1, 264 computations are independent of each other. We split it into 10 jobs. To deal with 120 representatives E 3 took between 5 and 10 hours on a Core i7 870 (2.93GHz) personal computer.
This computation shows the following. 3.3. The computations for D 8 . For this section we assume that H ∼ = D 8 . Then π 1 (G) = G 36 and we may use Lemma 1 to compute a system of representatives of the π 1 (G)−orbits on the set D.
Remark 8. π 1 (G) acts on D with exactly 9, 590 orbits represented by, say, X 1 , . . . , X 9,590 . For all 1 ≤ i ≤ 9, 590 we compute the code E := E(X i , σ) :=D +D σ , whereD = π −1 1 (X i ). For four X i the code E(X i , σ) is doubly even and of minimum distance 16. Let us call the inequivalent codes E 1 , E 2 , E 3 and E 4 , respectively. All have dimension 26 and minimum distance 16. Corollary 2. The code C(g) + C(h) + C(gh) is equivalent under the action of G to one of the four codes E 1 , E 2 , E 3 or E 4 . This computation is very fast (it is due mainly to the fact that G 36 = π(G)). It took about 5 minutes CPU on an Intel(R) Xeon(R) CPU X5460 @ 3.16GHz.
As it seems to be quite hard to compute all D 8 -invariant self-dual overcodes of E i for these four codes E i we apply a different strategy which is based on the fact that h = (gσ) 2 is the square of an element of order 4. So let k := gσ = (1, 8, 3, 6)(2, 5, 4, 7) . . . (66, 69, 68, 71) ∈ D 8 .
By [11], C is a free F 2 k -module (of rank 9). Since k is abelian, the module is both left and right; here we use the right notation. The regular module F 2 k has a unique irreducible module, 1-dimensional, called the socle, that is (1+k+k 2 +k 3 ) . So C, as a free F 2 k -module, has socle C(k) = C ·(1+k+k 2 +k 3 ). This implies that, for every basis b 1 , . . . , b 9 of C(k), there exist w 1 , . . . , w 9 ∈ C such that w i ·(1+k +k 2 +k 3 ) = b i and C = w 1 · F 2 k ⊕ . . . ⊕ w 9 · F 2 k . To get all the possible overcodes of E i , we choose a basis of the socle E i (k), say b 1 , . . . , b 9 , and look at the sets i /E i | w · (1 + k + k 2 + k 3 ) = b j and d(E i + w · F 2 k ) ≥ 16} For every i we have at least one j for which the set W i,j is empty. This computation (of about 4 minutes CPU on the same computer) shows the following.