Automorphisms of doubly-even self-dual binary codes

Submitted exclusiv ely to the London Mathematica l So ciet y doi:10.1112/0000 /000000 Automo rphisms of doubly-even self-dual bi nary co des. Annik a G ¨ unt her and Gabriele Neb e Abstract The automorphism group of a binary doubly-even self-dual code is alw a ys contained in the alternating group. O n the other hand, given a p ermutatio n group G of degree n there exists a doubly-even self-dual G -inv arian t code if an d only if n is a multiple of 8, every simple self-dual F 2 G -mod ule o ccurs with even multipli city in F n 2 , an d G is contained in the alternating group. 1. Int ro duction. Self-dual bina r y co des hav e b ecome o f great interest, also becaus e of Glea son’s theorem [ 6 ] that establis he s a connection be t ween co ding theor y and inv ariant theor y of finite g roups. Optimal self-dual co des o ften hav e the additional prope r t y of b eing doubly-even, which means that the w eight of every co dew o r d is divisible b y 4 (see Definition 1). It follows from Gleason’s theorem that the length n of a doubly-even self-dual co de C ≤ F n 2 is a m ultiple o f 8, see [ 9 , Theorem 3c], for instance. This note studies the automorphism group Aut( C ) := { π ∈ Sym n | C π = C } of such a code. Theorem 5.1 shows that the automor phis m group of any doubly-even self-dual co de is a lw ays contained in the a lternating group, a very bas ic res ult which a stonishingly do es not seem to be known. On the other hand Theorem 5.2 c haracteriz e s the per m utation gr oups G ≤ Sym n that fix a doubly-even se lf-dual binary co de. This result gener alizes results by Sloane a nd Thompson [ 13 ] and Mart ´ ınez-P´ erez and Willems [ 15 ]. The first section considers co des as mo dules fo r their automorphism gr oup. The main result is the characterization of p ermutation g roups that act o n a self-dual co de in Theorem 2.1. Section 4 treats p ermutation gr oups as subgroups of the 2 -adic ortho gonal groups . The most impo rtan t obser v a tion is Lemma 4.2 that expr e sses the sign of a p erm utation as a certain spinor no rm. Giv en a s elf-dual doubly-even binary co de C , the a utomorphism gro up o f the even unimo dular Z 2 -lattice obtained from C b y co ns truction A (see Section 3) is contained in the kernel o f this spinor norm. This immedia tely yields Theor em 5.1. Theorem 5.2 fo llo ws from this result together with Theor em 2.1. 2. Co des. Definition 1 . A binary co de C of length n is a linear s ubspace of F n 2 . Let b : F n 2 × F n 2 → F 2 , b ( x, y ) := P n i =1 x i y i be the standard scalar pro duct. The dual co de is C ⊥ := { v ∈ F n 2 | b ( v , c ) = 0 for all c ∈ C } . The code C is called self-orthog onal if C ⊆ C ⊥ and se lf-dua l if C = C ⊥ . The weight wt( c ) of a co deword c ∈ C is the num ber of its no nzero en tr ies. The code C is called doubly-even , or Type I I , if the weigh t of every w or d in C is a mu ltiple of 4. 2000 Mathematics Sub j ect Classi fication 94B05 (primary), 20G25, 11E95 (secondary). Page 2 o f 1 0 ANNIKA G ¨ UNTHER AND GABRIELE NEBE This section inv estiga tes bin ary linear co des as modules for a subgroup G of their automorphism g roups. The main res ult is Theorem 2.1 that characterizes the p ermutation groups acting on some s elf-dual code. T o this aim w e need the repr esen tation theor etic notion of self-dual mo dules, cf. Definition 2. Note that this pap er uses tw o differe nt notions of duality . The dual of an F 2 G -mo dule S over the finite group G is the F 2 G -mo dule S ∗ = Ho m F 2 ( S, F 2 ), whereas the dual o f a co de C ≤ F n 2 is a s in Definition 1. F or G ≤ Aut( C ) the co de C is also an F 2 G -mo dule, whic h is represented with re spect to a distinguished ba sis. Definition 2. Let S be a righ t G -module. Then the dual module S ∗ = Ho m F 2 ( S, F 2 ) is a right G - mo dule via f g ( s ) := f ( sg − 1 ), for f ∈ S ∗ , g ∈ G and s ∈ S . If S ∼ = S ∗ then S is called self-dual . Theorem 2.1. Let G ≤ Sym n . Then there exists a self-dual co de C ≤ F n 2 with G ≤ Aut( C ) if and only if ev er y self-dua l simple F 2 G -mo dule S o ccurs in the F 2 G -mo dule F n 2 with even m ultiplicit y . The pro of of this theorem is prepared in a few lemmas. Lemma 2.2. Let S b e a s imple self-dual F 2 G -mo dule, and assume that S carrie s a no n- degenerate symmetric G -inv ariant bilinear form ϕ : S × S → F 2 . Then ϕ is unique , up to isometry . Pro of. Since ϕ is no n-degenerate and G -inv ariant, it induces a n F 2 G -isomorphism α ϕ : S → S ∗ , s 7→ ( s ′ 7→ ϕ ( s, s ′ )). Let ψ : S × S → F 2 be another non-deg enerate s y mmetric G -inv ariant bilinear form on S , then α ψ = α ϕ ◦ ϑ for some ϑ in the finite field E := End G ( S ) of a ll F 2 G - endomorphisms of S , a nd hence ψ ( s, s ′ ) = α ψ ( s )( s ′ ) = α ϕ ( ϑ ( s ))( s ′ ) = ϕ ( ϑ ( s ) , s ′ ) for a ll s, s ′ ∈ S . Co nsider the involution ad on E given by ϕ ( s, α ( s ′ )) = ϕ ( α ad ( s ) , s ′ ), f or s, s ′ ∈ S . Since b oth ϕ and ψ are symmetric we ha ve ϕ ( ϑ ( s ) , s ′ ) = ψ ( s, s ′ ) = ψ ( s ′ , s ) = ϕ ( ϑ ( s ′ ) , s ) = ϕ ( s, ϑ ( s ′ )) = ϕ ( ϑ ad ( s ) , s ′ ) for a ll s, s ′ ∈ S and hence ϑ ∈ F = { α ∈ E | α ad = α } . The inv o lutio n ad is either the identit y on E or a field automo rphism of order 2. In the first case F = E = { αα ad = α 2 | α ∈ E } since squaring is an auto morphism of the finite fie ld E . In the second case the ma p E → F , α 7→ α α ad is the norm ma p onto the fixed field F . Hence in either case there exists some γ ∈ E with γ γ ad = ϑ . Now γ induces an isometry b etw een the spaces ( S, ϕ ) and ( S, ψ ) since ψ ( s, s ′ ) = ϕ ( ϑ ( s ) , s ′ ) = ϕ ( γ ad ( γ ( s )) , s ′ ) = ϕ ( γ ( s ) , γ ( s ′ )) for all s, s ′ ∈ S . Lemma 2.3. Let G ≤ Sym n and let N ≤ M ≤ F n 2 be G -submo dules (i.e. G -inv ar ian t co des). Then ( M / N ) ∗ ∼ = N ⊥ / M ⊥ . Pro of. Let M ∗ N := { f ∈ Ho m F 2 ( M , F 2 ) | f ( n ) = 0 for all n ∈ N } ≤ M ∗ . Then M ∗ N is canonically isomorphic to ( M / N ) ∗ . Let β : N ⊥ → M ∗ N , n ′ 7→ ( m 7→ b ( m, n ′ )). Then β is well- defined and s urjectiv e , since Υ : F n 2 → M ∗ , v 7→ ( m 7→ b ( m, v )) is surjective, and Υ( v ) ∈ M ∗ N if and only if v ∈ N ⊥ . Clearly β has kernel M ⊥ and hence N ⊥ / M ⊥ ∼ = M ∗ N ∼ = ( M / N ) ∗ . AUTOMORPHISMS OF DOUBL Y-EVEN SELF-DUAL BINAR Y CODES. P age 3 of 10 Corollar y 2.4. Let G ≤ Sym n . If there exists a self-dual co de C ≤ F n 2 with G ≤ Aut( C ) then ev ery self-dual simple G -mo dule o ccurs with even mu ltiplicit y in a co mposition series of the F 2 G -mo dule F n 2 . Pro of. Let C = N k ≥ N k − 1 ≥ . . . ≥ N 1 ≥ N 0 = { 0 } b e a comp osition series of the F 2 G - mo dule C . Then C = C ⊥ = N ⊥ k ≤ N ⊥ k − 1 ≤ . . . ≤ N ⊥ 1 ≤ N ⊥ 0 = F n 2 is a comp osition se r ies o f F n 2 /C ⊥ , since dualizing yields an an tiautomorphism W 7→ W ⊥ of the submo dule lattice of F n 2 . The comp osition factors satisfy N ⊥ i − 1 / N ⊥ i ∼ = ( N i / N i − 1 ) ∗ , cf. Lemma 2.3. Hence the claim follows. Lemma 2.5. Let S b e a simple s elf-dual F 2 G -mo dule endo wed with a non-deg enerate G - inv aria n t sy mmetr ic bilinear form ϕ . The module ( U, ψ ) := ⊥ k i =1 ( S, ϕ ) co n tains a submodule X with X = X ⊥ ,ψ := { u ∈ U | ψ ( u, x ) = 0 for all x ∈ X } if and only if k is even. Pro of. If U co n tains such a submo dule X = X ⊥ ,ψ then k is even accor ding to Corollary 2.4. Con versely , if k is ev en then X := { ( s 1 , s 1 , s 2 , s 2 , . . . , s k/ 2 , s k/ 2 ) } ≤ U sa tis fie s X = X ⊥ ,ψ . Pro of. (of Theor em 2.1) If C ≤ F n 2 =: V is a self-dual G -inv ariant co de then every self-dual simple module o ccurs with ev en m ultiplicity in a comp osition series o f V (see Corollar y 2.4). Conv er s ely , ass ume that ev ery self-dual composition factor occurs in V with even multiplicit y , and let M ≤ M ⊥ ≤ V be a maximally self-orthogona l G -inv a riant code, i.e. there is no self- orthogo nal G -inv ariant co de in V which pro perly contains M . On the G -mo dule M ⊥ / M ther e exists a G -inv ariant non- degenerate symmetric bilinear for m ϕ : M ⊥ / M × M ⊥ / M → F 2 , ( m ′ + M , m ′′ + M ) 7→ ( m ′ , m ′′ ) . An y prop er F 2 G -submo dule X of ( M ⊥ / M , ϕ ) with X ⊆ X ⊥ ,ϕ (cf. Lemma 2.5) would lift to a self-or thogonal G -inv ariant co de in V pro p erly containing M , which we excluded in our ass umptions. T his implies that every F 2 G -submo dule X ≤ M ⊥ / M has a G -in v ariant complement X ⊥ ,ϕ , i.e. M ⊥ / M is isomor phic to a direct sum of simple self-dual mo dules (see for instance [ 3 , Prop osition (3.1 2)]), ( M ⊥ / M , ϕ ) ∼ = ⊥ S ∼ = S ∗ ( S, ϕ S ) n S , where ϕ S is a no n-degenerate G -inv aria n t bilinear form on S , whic h is unique up to isometry b y Lemma 2.2. According to our ass umptions, every simple self-dua l G -mo dule o ccurs with even multiplicit y in M ⊥ / M , i.e. all the n S are even. But this means that the n S m ust all b e zero, according to Lemma 2.5, that is, M = M ⊥ is a self-dual co de in V . The criterion in Theorem 2.1 is not so easily tested. The next result gives a g roup th eoretic condition that is sufficient for the ex istence of a se lf-dua l G -in v ariant code. T o this aim let G ≤ Sym n be a permutation group and write { 1 , . . . , n } = B 1 . ∪ . . . . ∪ B s Page 4 o f 1 0 ANNIKA G ¨ UNTHER AND GABRIELE NEBE as a disjoin t union of G -orbits and let H i := Sta b G ( x i ) be the sta bilizer in G of so me elemen t x i ∈ B i ( i = 1 , . . . , s ). F or 1 ≤ i ≤ s let m i := |{ j ∈ { 1 , . . . , s } | H i is conjugate to H j }| and n i := [ N G ( H i ) : H i ] . Proposition 2.6 . Assume that the pro duct n i m i is even for all 1 ≤ i ≤ s . Then there is a G -in v ariant self- dua l binary code C ≤ F n 2 . Pro of. If H i and H j are conjugate for s ome i 6 = j then the p e rm utatio n representations o f G on B i and B j are equiv a len t and by T he o rem 2.1 there is a self-dua l G -inv ar ia n t co de in the direct sum F B i ∪ B j 2 ∼ = F | B i | 2 ⊥ F | B j | 2 of tw o isomorphic F 2 G -mo dules. It is hence enough to show the pro position for a tr ansitiv e p erm utation g roup G ≤ Sym n with stabilizer H := Stab G (1) for which [ N G ( H ) : H ] ∈ 2 Z . Let ( f 1 , . . . , f n ) b e the standard basis o f F n 2 such that π ∈ Sym n maps f j to f j π for all j = 1 , . . . , n and cho ose η ∈ N G ( H ) − H s uc h that η 2 ∈ H . Put N := h H , η i and G = . ∪ s ∈ S N s = . ∪ s ∈ S ( H s . ∪ H η s ) . Define C := h f 1 s + f 1 ηs : s ∈ S i F 2 . Then C is a G -in v ariant code in F n 2 and C = C ⊥ since the given basis of C cons is ts of | S | = n/ 2 pairwise or thogonal vectors of weigh t 2. 3. F rom co des to lattices. There is a well-known co nstruction, called construction A (see [ 2 , Section (7.2)]) that asso ciates to a pair ( R, C ) of a ring R with prime ideal ℘ and residue field R/℘ ∼ = F and a co de C ≤ F n an n -dimensional lattice over R . W e w ill apply this construction for binary co des a nd tw o different base rings: R = Z a nd R = Z 2 , the ring of 2-adic in tegers, where the prime ideal ℘ = 2 R in b oth cases. So let R b e one of these t wo rings and let K denote the field of fractions of R and let V := h b 1 , . . . , b n i K be a vector space ov er K with bilinear form defined by ( , ) : V × V → K , ( b i , b j ) := 1 2 δ ij =  1 / 2 i = j 0 i 6 = j and asso ciated quadratic form q : V → K , q ( v ) := 1 2 ( v , v ). The orthogona l group of V is O ( V ) := { g ∈ GL( V ) | ( v g , wg ) = ( v , w ) for all v , w ∈ V } . Definition 3. A lattice L ≤ V is the R -span of a basis of V . The dual lattice L # := { v ∈ V | ( v , ℓ ) ∈ R for all ℓ ∈ L } is again a lattice in V . L is ca lled in tegr al if L ⊆ L # or equiv alently ( ℓ 1 , ℓ 2 ) ∈ R fo r all ℓ 1 , ℓ 2 ∈ L . L is ca lle d even if q ( ℓ ) ∈ R for all ℓ ∈ L a nd odd if L is int egral and there is some ℓ ∈ L with q ( ℓ ) 6∈ R . L is called unimo dular if L = L # . The orthog onal gr oup of L is O ( L ) := { g ∈ O ( V ) | Lg = L } . The following remar k lists elementary pro perties of the lattice o btained fro m a co de by construction A which can be seen by straightf orward calculations . AUTOMORPHISMS OF DOUBL Y-EVEN SELF-DUAL BINAR Y CODES. P age 5 of 10 Remark 1. Let M = h b 1 , . . . , b n i R be the lattice g enerated by the basis a bov e and let C ≤ F n 2 be a binary co de. Then the R -lattice L := A ( R, C ) := { n X i =1 a i b i | a i ∈ R, ( a 1 + 2 R, . . . , a n + 2 R ) ∈ C } is ca lled the co delattice o f C . Note tha t 2 M ⊂ L ⊂ M a nd L is the f ull preimage of C ∼ = L/ 2 M under the na tural epimo rphism M → ( R/ 2 R ) n = F n 2 . The la ttice L is even if and only if the co de C is doubly-ev en. The dual lattice is A ( R, C ) # = A ( R , C ⊥ ) and hence L is unim o dular if and o nly if C is self-dual, and L is an e ven unimo dular lattice if and only if C is a doubly-ev en self-dual co de. The sy mmetric gro up Sym n acts as or thogonal transforma tio ns o n V by p erm uting the ba s is vectors. This yields an injective homo morphism ι : Sym n → O ( V ) , ι ( π ) : b i 7→ b iπ . If G = Aut( C ) is the automor phism gr oup of C then ι ( G ) ≤ O ( A ( R, C )). 4. P ermutations as elements of the orthogona l group. Let Q 2 denote the field of 2-adic num ber s, v 2 : Q 2 → Z ∪ {∞} its na tural v aluation and Z 2 := { x ∈ Q 2 | v 2 ( x ) ≥ 0 } the ring of 2-adic in teger s with unit gr oup Z ∗ 2 := { x ∈ Z 2 | v 2 ( x ) = 0 } . Let V := h b 1 , . . . , b n i Q 2 be a bilinear spa c e ov er Q 2 of dimensio n n > 1 as in Section 3, in particular ( b i , b j ) = 1 2 δ ij . The orthogona l gr oup O ( V ) is generated by a ll reflections σ v : V → V , x 7→ x − ( x, v ) q ( v ) v along v ec tors v ∈ V with q ( v ) 6 = 0 (see [ 7 , Satz (3.5)], [ 10 , Theorem 43:3 ]). Then the spinor norm defines a gro up homomor phism h : O ( V ) → C 2 as follows: Definition 4. Let h : O ( V ) → C 2 = { 1 , − 1 } b e defined by h ( σ v ) := ( − 1) v 2 ( q ( v )) for all reflections σ v ∈ O ( V ). Let O h ( V ) := { g ∈ O ( V ) | h ( g ) = 1 } deno te the kernel of this epimorphism. Note that the definition of h depends on the chosen scaling of the quadratic for m. It follo ws from the definition of the spinor no rm (see [ 10 , Sec tion 55]) that Lemma 4.1. The map h is a well-defined group epimorphism. The crucial observ a tion that yields the co nnection to co ding theory in Section 5 is the following easy le mma . Lemma 4.2. Let ι : Sym n → O ( V ) b e the homomorphism from Remar k 1. Then h ◦ ι = sign . Pro of. The symmetric gr oup Sym n is g enerated by transp ositions τ i,j = ( i, j ) for i 6 = j . Such a transp osition interc hanges b i and b j and fixes all other basis vectors and hence ι ( τ i,j ) = σ b i − b j . Clearly h ( σ b i − b j ) = ( − 1) v 2 ( q ( b i )+ q ( b j )) = ( − 1) − 1 = − 1 = sign( τ i,j ) . Page 6 o f 1 0 ANNIKA G ¨ UNTHER AND GABRIELE NEBE Lemma 4.3. Let L ≤ V b e an even unimodular lattice. Then O ( L ) ≤ O h ( V ) . Pro of. By [ 7 , Satz 4.6 ] the o r thogonal g roup O ( L ) is gener ated b y reflections O ( L ) = h σ ℓ | ℓ ∈ L, v 2 ( q ( ℓ )) = 0 i . Since h ( σ ℓ ) = ( − 1) v 2 ( q ( ℓ )) = 1 for those vectors ℓ , the result follows. W e now assume tha t n is a m ultiple of 8 a nd c ho ose an orthonor ma l basis ( e 1 , . . . , e n ) of V (i.e. ( e i , e j ) = δ ij ). Let L := h e 1 , . . . , e n i Z 2 be the unimo dular lattice g enerated by these vectors e i and let L 0 := { ℓ ∈ L | q ( ℓ ) ∈ Z 2 } = h e 1 + e 2 , . . . , e 1 + e n , 2 e 1 i be its even sublattice. Then L # 0 = h e 1 , . . . , e n − 1 , v := 1 2 P n i =1 e i i . Since n is a m ultiple of 8 the vector 2 v ∈ L 0 and ( v , v ) = n 4 is even. Hence L # 0 /L 0 ∼ = F 2 2 and the three lattices L i with L 0 < L i < L # 0 corres p onding to the three 1-dimensional subspaces of L # 0 /L 0 are given by L 1 := h L 0 , v i , L 2 := h L 0 , v − e 1 i , L 3 = L . Note that L 1 and L 2 are even unimo dular lattices, whereas L 3 is o dd. In particular O ( L ) = O ( L 0 ) acts as the subgro up { 1 , − 1 } = C 2 ∼ = { I 2 ,  1 1 0 1  } ≤ GL 2 ( F 2 ) on L # 0 /L 0 (with resp ect to the basis ( v + L 0 , e 1 + L 0 )). Let f : O ( L ) → C 2 = {± 1 } denote the resulting epimorphism. So the elements in the kernel of f (whic h equals O ( L ) ∩ O h ( V ) as shown in the next lemma ) fix both lattices L 1 and L 2 and all other elemen ts in O ( L ) in ter change L 1 and L 2 . Lemma 4.4. f = h | O ( L ) Pro of. Let R ( L 0 ) := h σ ℓ | ℓ ∈ L 0 , q ( ℓ ) ∈ Z ∗ 2 i b e the reflection subgroup of O ( L 0 ). By [ 8 , Satz 6] R ( L 0 ) is the kernel o f f . Since h ( σ ℓ ) = 1 for all σ ℓ ∈ R ( L 0 ), the g roup R ( L 0 ) ⊂ O ( L ) ∩ O h ( V ) is also contained in the kernel of h . The reflectio n σ e 1 along the vector e 1 ∈ L is in the o r thogonal group O ( L ) = O ( L 0 ), int erchanges the tw o lattices L 1 and L 2 , and satisfies h ( σ e 1 ) = − 1 . Since R ( L 0 ) is a nor mal subgroup of index at mos t 2 in O ( L ), we obtain O ( L ) = h R ( L 0 ) , σ e 1 i and the lemma follows. 5. The main results. Theorem 5.1. Let C = C ⊥ ≤ F n 2 be a do ubly - ev e n self- dua l co de. Then the automorphis m group of C is con tained in the alternating group. Pro of. W e apply construction A fro m Sectio n 3 to the co de C to obta in the co delattice L := A ( Z 2 , C ). By Remark 1 the lattice L is an ev en unimo dular lattice. Hence b y Lemma 4.3 its orthogo nal gro up O ( L ) ≤ O h ( V ) is in the kernel of the epimorphism h from Definit ion 4. The image of Aut( C ) under the homomorphism ι from Remark 1 is con tained in O ( L ), henc e AUTOMORPHISMS OF DOUBL Y-EVEN SELF-DUAL BINAR Y CODES. P age 7 of 10 ι (Aut( C )) ≤ O ( L ) ≤ O h ( V ). Since h ◦ ι = sign by Lemma 4.2 we have s ign(Aut( C )) = { 1 } and therefore Aut( C ) ≤ Alt n . Theorem 5.2 . Let G ≤ Sym n . Then there is a s elf-dual doubly-even co de C = C ⊥ ≤ F n 2 with G ≤ Aut( C ) if and only if the following three conditions are fulfilled: (a) 8 | n . (b) Every self-dual compositio n factor o f the F 2 G -mo dule F n 2 o ccurs with ev en m ultiplicit y . (c) G ≤ Alt n . Pro of. ⇒ : (a) is clear since the leng th o f a n y doubly-even se lf- dua l code is a m ultiple of 8. (b) follows f rom Theorem 2.1 and (c) is a cons equence of Theorem 5.1. ⇐ : By Theorem 2.1 the co ndition (b) implies the existence of a se lf- dua l co de X = X ⊥ with G ≤ Aut( X ). If X is doubly-even then we ar e done. So assume that X is not doubly-ev en and consider the co delattices L := A ( Z , X ) and L X := A ( Z 2 , X ) = L ⊗ Z 2 . Then L is a p ositive definite o dd unimo dular Z -la ttice and hence its 2-adic completion L ⊗ Z 2 = L X is an o dd unimo dular Z 2 -lattice having an orthono rmal basis (see for ins ta nce [ 7 , Satz (26.7)]). Hence L X is isometric to the lattice L co nstructed just b e fore Le mma 4.4. Since G ≤ Alt n , the gr o up ι ( G ) ≤ O ( L X ) lies in the kernel of the homomo rphism f from Lemma 4.4 and therefore fixes the tw o even unimo dular lattices L 1 and L 2 int ersecting L X in its even sublattice. Let M = h b 1 , . . . , b n i Z 2 be the la ttice from Remar k 1 such that 2 M < L X < M and identify M / 2 M = L n i =1 Z 2 / 2 Z 2 b i = L n i =1 F 2 b i with F n 2 . Then the co de C := L 1 / 2 M ≤ F n 2 (such that L 1 = A ( Z 2 , C )) is a self-dual doubly-even co de with G ≤ Aut( C ). 6. An application to group ring co des. As an application of our main Theorem 5 .2 we obtain a r esult (Theorem 6.3) o n the existence of self-dua l doubly-even bina ry gro up co des, giv en in [ 13 ] and also in [ 15 ]. Binary group co des are ideals of the gr oup ring F 2 G , where G is a finite gro up, i.e. these ar e exactly the co des in F | G | 2 with ρ G ( G ) ≤ Aut( C ), where ρ G : G → Sym G , g 7→ ( h 7→ hg ) is the reg ular repr esen tation of G . C le arly ρ G ( G ) ≤ Alt G if and only if the image ρ G ( S ) of a n y Sylow 2- subgroup S ∈ Syl 2 ( G ) is contained in the alter nating gro up. Let k := [ G : S ] b e the index of S in G . Then k is o dd and the restriction of ρ G to S is ( ρ G ) | S = k ρ S . Hence ρ G ( S ) ≤ Alt G if and only if ρ S ( S ) ≤ Alt S . Lemma 6.1. Let S 6 = 1 b e a 2-gro up. Then ρ S ( S ) ≤ Alt S if and only if S is no t cyclic. Pro of. If S = h s i is cy c lic, then ρ S ( s ) is a | S | - cycle in Sym S and hence its sig n is -1 (b ecause | S | is ev en). On the other hand assume that S is not cyclic. Then S ha s a normal subgr oup N such tha t S/ N ∼ = C 2 × C 2 is g enerated by elements aN , bN ∈ S/ N of order 2, with abN = baN . Let A := h a, N i and B = h b, N i . Then S = h A, B i = A . ∪ bA = B . ∪ aB and b induces an isomorphism b et ween the regular A -mo dule A and bA , so A is in the k ernel of the sign homomorphism. Similar ly a gives an isomo r phism b et ween the regula r B -module B and aB , so also B is in the kernel of the sign homomorphism. The following o bserv ation follows from Pr o position 2.6 and is proven in [ 14 , Theorem 1.1 ]. Page 8 o f 1 0 ANNIKA G ¨ UNTHER AND GABRIELE NEBE Theorem 6.2. There is a self-dual binary group co de C ≤ F 2 G if and only if the o rder of G is even. Pro of. ⇒ : Clear, since dim( C ) = | G | 2 for any C = C ⊥ ≤ F 2 G . ⇐ : F ollows from Prop osition 2.6, because ρ G is a transitive p ermut ation representation and the the full group G is the norma lizer o f the stabilizer H := Stab G (1) = 1. Theorem 6.3. ( see [ 13 ],[ 15 ]. ) Let G be a finite gr oup. Then F 2 G contains a doubly- e v en self-dual group co de if and only if the order o f G is divisible b y 8 and the Sylo w 2 - subgroups of G are not cyclic. Pro of. The condition that the group order be divisible by 8 is e quiv a len t to condition (a) o f Theorem 5.2 and als o implies (with Theorem 6.2) tha t there is some self-dual G - inv ariant co de in F 2 G , which is equiv alent to co ndition (b) o f Theorem 5.2 by Theorem 2.1. The condition on the Sylow 2-subgroups o f G is equiv alent to ρ G ( G ) ≤ Alt G by Lemma 6.1 and he nc e to condition (c) of Theorem 5.2. Our last application co ncerns the automo rphism group G = Aut( C ) of a putative extr emal Type I I co de C of length 72. The pa per [ 1 ] shows that any automorphism of C of o rder 2 acts fixed p oint freely , so any Sylow 2-subgroup S of G acts as a m ultiple of the re gular representation. In pa rticular | S | divides 8. Our results show that S is no t cy clic of order 8, which alr eady fo llo ws fro m [ 1 3 , Theorem 1]. Corollar y 6.4. Let C b e a s elf-dual doubly-even binar y c o de of length 72 with minimum distance 16. Then C does not ha ve an automorphism of order 8. 7. A c ha racteristic 2 proo f o f Theorems 5.1 and 5.2 As remarked b y Rob ert Griess one ma y pro ve Theo rem 5.1 and 5.2 without using characteristic 0 theor y . Assume that n is a multiple of 8 and let 1 := (1 , . . . , 1) ∈ F n 2 denote the all ones vector. Then V = 1 ⊥ / h 1 i = { x ∈ F n 2 | wt( x ) is ev en } / h 1 i bec omes a quadratic module of dimension n − 2 over F 2 by putting q : V → F 2 , x := x + h 1 i 7→ 1 2 wt( x ) + 2 Z . The a ssocia ted bilinea r form b ( x, y ) = q ( x + y ) − q ( x ) − q ( y ) = x · y is inherited from the standard inner pro duct and the maximal isotropic subspaces of V are the images of the doubly-even self- dua l codes in F n 2 . The orthogonal g r oup O ( V , q ) ∼ = O + n − 2 (2) acts tra nsitiv ely on the set of maximal is o tropic subspaces of V . Fix one such subspace U . Then the Dic kso n in v aria n t is D : O ( V , q ) → { 1 , − 1 } ; D ( g ) := ( − 1 ) dim( U /U ∩ U g ) a well-defined homomorphism that do es not depend on the choice of U ([ 12 , Theorem 11 .61]). The symmetric gr oup Sym n acts by coo rdinate p ermutations on F n 2 . Since 1 π = 1 for all π ∈ Sym n and p ermutations preserve the weight this gives rise to an embedding ι : Sym n → O ( V , q ). The following lemma also follows from the geometric c haracteriza tion of the Dickson inv aria n t in [ 12 , p. 160] (see also [ 5 ] and [ 4 ]). AUTOMORPHISMS OF DOUBL Y-EVEN SELF-DUAL BINAR Y CODES. P age 9 of 10 Lemma 7.1. D ◦ ι = sign . Pro of. It is e nough to find a transp osition that is not in the kernel of the Dickson inv ariant. T o this aim cho o se the Typ e II co de C with generator matrix        1 1 1 1 0 0 0 . . . 0 0 0 1 1 0 0 1 1 0 . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 0 0 0 0 0 . . . 0 1 1 1 0 1 0 1 0 . . . . . . . . . 1 0        and let U := C / h 1 i . Then U ι ( τ 1 , 2 ) ∩ U has co-dimensio n 1 in U. Now we can use the Dickson inv ariant D to replace the spinor norm h to obtain the main results. It is immediate that Stab O ( V , q ) ( U ) ⊂ ker ( D ) (see also [ 12 , Ex ercise 1 1 .19]) fro m whic h one obtains Theorem 5.1. The pr oof o f Theorem 5.2 ca n also b e mo dified. Co ndition (b) implies the exis tence of a self-dual G -inv ar ian t co de X . If X is doubly- ev e n, then we are done; if not, then let X 0 := { x ∈ X | wt( x ) ∈ 4 Z } denote the doubly-even sub code of X . This is a sub code of codimension 1 in X and X ⊥ 0 /X 0 ∼ = F 2 ⊕ F 2 is of dimens io n 2. Since the length of X is divisible b y 8 , the full pr eimages C 1 and C 2 of the other tw o non-trivia l subspaces of X ⊥ 0 /X 0 bo th are self-dua l doubly-even co des. Since the co-dimension o f the intersection dim( C i / ( C 1 ∩ C 2 )) = 1 is o dd, any p erm utation π with C 1 π = C 2 has to ha ve sign( π ) = D ( ι ( π )) = − 1. Since G ≤ Alt n , all elements o f G ha ve to fix b oth co des C 1 and C 2 and hence these yield G -inv ariant doubly-even self-dual co des. The pro of of Theor em 5.1 given here directly gener alizes to gener alized doubly-even co des as well a s to odd c haracter istic. Note that in odd c haracteristic the Dickson inv ar ian t is the same as the determinan t of an or tho gonal mapping. F or further details we r efer to the first author’s thesis. Theorem 7 .2. (a) Let C = C ⊥ ≤ F n 2 d be a gene r alized doubly even co de as defined in [ 11 ]. Then P ( C ) ≤ Alt n . (b) Let q b e an odd prime p o wer and C = C ⊥ = { x ∈ F n q | P n i =1 x i c i = 0 for a ll c ∈ C } . Then any mo nomial automorphism g ∈ Stab C 2 ≀ S n ( C ) has determinan t 1. References 1. S. Bouyukliev a, On the automorphisms of order 2 with fixed points for the extremal self-dual codes of length 24 m . Designs, Co des, Cryptogr. 25 , 5- 13 (2002) 2. J. H. Conw ay , N.J.A . Sloane, Sphere pa c kings, lattices and groups. Springer Grundlehren 290, 1993. 3. C. W. Curtis, I. Reiner, Methods of Represen tation Theory I. Wiley classics 1990. 4. J. Dieudonn´ e, Pseudo-discrim inan t and Dick son inv ar i an t. Pacific J. M ath. 5 , 907-910 (1955) 5. R . H. Dye, A geometric charact erization of the sp ecial orthogonal groups and the Dickson inv ari an t. J. LMS (2) 15 , 472-476 (1977) 6. A . M . Gleason, W eigh t p olynomials of self-dual co des and the M acWill iams i den tities. in Actes, Congr´ es In ternational de M ath ´ ematique s (Nice, 1970) , G authiers-Villars, Paris, 1971, V ol. 3, pp. 211–215. 7. M . Kneser, R. Sch arlau, Quadratische F ormen. Springer 2002 8. M . Kneser, Erzeugung ganzzahliger orthog onaler Gruppen durch Spiegelungen. Mathem. Annalen 255, 453-462 (1981) 9. F. J. MacWilliams, N.J.A . Sloane, The Theory of Error-Correcting Co des , North-Holland, Am sterdam, 1977; 11th i mpression 2003. 10. O.T. O’ Meara, Int ro duct ion t o Quadratic F orms . Springer Grundlehren 117, 1973. 11. H.-G. Quebb emann, On even codes, Discrete Math. 98 (1991), no. 1, 29–34. 12. C.E. T ay lor, The geometry of the classical groups. Heldermann V erlag Berlin 1992. Page 10 of 1 0 AUTOMORPHISMS OF DOUBL Y-EVEN SELF-DUAL BINAR Y CODE S. 13. N.J. A . Sloane, J.G. Thompson, Cyclic Self-Dual Codes. I EEE T rans. Inform. Theory 29, 1983 14. W. Will ems, A not e on s elf-dual group co des. IEEE T rans. Inform. Theory 48 (2002), no. 12, 3107–3109. 15. C. Mart ´ ınez-P´ erez, W. Willems, Self-dual co des and mo dules for finite groups i n c haracteristic tw o. IEEE T r ans. Inform. Theory 50 (2004), no. 8, 1798–1803. A. G¨ un ther and G. Neb e, Lehrstuhl D f¨ ur Mathematik, R WTH Aac hen Universit y 52056 Aac hen, Germany annik a.guenther@math.rwth-aachen.de, gabriele.neb e@math.rwth-aa c hen.de