# On Extremal Self-Dual Ternary Codes of Length 48

Authors: Gabriele Nebe
Journal: International Journal of Combinatorics
 (2012)

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## Abstract

All extremal ternary self-dual codes of length 48 that have some automorphism of prime orderp≥5 are equivalent to one of the two known codes, the Pless code or the extended quadratic residue code.


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## Body

## 1. Introduction

The notion of an extremal self-dual code has been introduced in [1]. As Gleason [2] remarks one may use invariance properties of the weight enumerator of a self-dual code to deduce upper bounds on the minimum distance. Extremal codes are self-dual codes that achieve these bounds. The most wanted extremal code is a binary self-dual doubly even code of length 72 and minimum distance 16. One frequently used strategy is to classify extremal codes with a given automorphism, see [3, 4] for the first papers on this subject.Ternary codes with a given automorphism have been studied in [5]. The minimum distance d(C):=min{wt(c)∣0≠c∈C} of a self-dual ternary code C=C⊥≤𝔽3n of length n is bounded by(1.1)d(C)≤3⌊n12⌋+3.
Codes achieving equality are called extremal. Of particular interest are extremal ternary codes of length a multiple of 12. There exists a unique extremal code of length 12 (the extended ternary Golay code), two extremal codes of length 24 (the extended quadratic residue code Q24:=QR̃(23,3) and the Pless code P24). For length 36, the Pless code yields one example of an extremal code. Reference [5] shows that this is the only code with an automorphism of prime order p≥5; a complete classification is yet unknown. The present paper investigates the extremal codes of length 48. There are two such codes known, the extended quadratic residue code Q48 and the Pless code P48. The computer calculations described in this paper show that these two codes are the only extremal ternary codes C of length 48 for which the order of the automorphism group is divisible by some prime p≥5. Theoretical arguments exclude all types of automorphisms that do not occur for the two known examples.Any extremal ternary self-dual code of length 48 defines an extremal even unimodular lattice of dimension 48 ([6]). A long-term project to find or even classify such lattices was my main motivation for this paper.


## 2. Automorphisms of Codes

Let𝔽 be some finite field, 𝔽* its multiplicative group. For any monomial transformation σ∈Monn(𝔽):=𝔽*≀Sn, the image π(σ)∈Sn is called the permutational part of σ. Then σ has a unique expression as(2.1)σ=diag⁡(α1,…,αn)π(σ)=m(σ)π(σ),
and m(σ) is called the monomial part of σ. For a code C≤𝔽n we let(2.2)Mon(C):={σ∈Monn(F)∣σ(C)=C}
be the full monomial automorphism group of C.We call a codeC≤𝔽n an orthogonal direct sum, if there are codes Ci≤𝔽ni (1≤i≤s>1) of length ni such that(2.3)C~⦹i=1sCi={(c1(1),…,cn1(1),…,c1(s),…,cns(s))∣c(i)∈Ci(1≤i≤s)}.Lemma 2.1.
LetC≤𝔽n not be an orthogonal direct sum. Then the kernel of the restriction of π to Mon(C) is isomorphic to 𝔽*.Proof.
Clearly𝔽*C=C since C is an 𝔽-subspace. Assume that σ:=diag(α1,…,αn)∈Mon(C) with αi∈𝔽*, not all equal. Let {α1,…,αn}={β1,…,βs} with pairwise distinct βi. Then
(2.4)C=⦹i=1sker⁡(σ-βiid)
is the direct sum of eigenspaces of σ. Moreover the standard basis is a basis of eigenvectors of σ so this is an orthogonal direct sum.In the investigation of possible automorphisms of codes, the following strategy has proved to be very fruitful ([4, 7]).Definition 2.2.
Letσ∈Mon(C) be an automorphism of C. Then π(σ)∈Sn is a direct product of disjoint cycles of lengths dividing the order of σ. In particular if the order of σ is some prime p, then we say that σ has cycle type (t,f), if π(σ) has t cycles of length p and f fixed points (so pt+f=n).Lemma 2.3.
Letσ∈Mon(C) have prime order p.(a)
Ifp does not divide |𝔽*| then there is some element τ∈Monn(𝔽) such that m(τστ-1)=id. Replacing C by τ(C) one hence may assume that m(σ)=1.(b)
Assume thatp does not divide char(𝔽), m(σ)=1, and π(σ)=(1,…,p)⋯((t-1)p+1,…,tp)(tp+1)⋯(n). Then C=C(σ)⊕E, where
(2.5)C(σ)={c∈C∣c1=⋯=cp,cp+1=⋯=c2p,…,c(t-1)p+1=⋯=ctp}
is the fixed code of σ and
(2.6)E={c∈C∣∑i=1pci=∑i=p+12pci=⋯=∑i=(t-1)p+1tpci=ctp+1=⋯=cn=0}
is the unique σ-invariant complement of C(σ) in C.(c)
Define two projections(2.7)πt:C(σ)⟶Ft,πt(c):=(cp,c2p,…,ctp),πf:C(σ)⟶Ff,πf(c):=(ctp+1,ctp+2,…,ctp+f).
So C(σ)≅(πt(C(σ)),πf(C(σ)))=:C(σ)*. If C=C⊥ is self-dual with respect to (x,y):=∑i=1nxiyi¯, then C(σ)*≤𝔽t+f is a self-dual code with respect to the inner product (x,y):=∑i=1tpxiyi¯+∑j=t+1t+fxjyj¯.(d)
In particulardim(C(σ))=(t+f)/2 and dim(E)=t(p-1)/2.Proof.
(a) follows from the Schur-Zassenhaus theorem in finite group theory. For the ternary case, see [5, Lemma  1].
(b) and (c) are similar to [4, Lemma  2].In the following we will keep the notation of the previous lemma and regard the fixed codeC(σ).Remark 2.4.
Iff≤d(C) then t≥f.Proof.
Otherwise the kernelK:=ker(πt)={(0,…,0,c1,…,cf)∈C(σ)} is a nontrivial subcode of minimum distance ≤f<d(C).The way to analyse the codeE from Lemma 2.3 is based on the following remark.Remark 2.5.
Letp≠char(𝔽) be some prime and σ∈Monn(𝔽) an element of order p. Let
(2.8)Xp-1=(X-1)g1⋯gm∈F[X]
be the factorization of Xp-1 into irreducible polynomials. Then all factors gi have the same degree d=|〈|𝔽|+pℤ〉|, the order of |𝔽| mod p.  There are polynomials ai∈𝔽[X] (0≤i≤m) such that
(2.9)1=a0g1⋯gm+(X-1)∑i=1mai∏j≠igj.
Then the primitive idempotents in 𝔽[X]/(Xp-1) are given by the classes of
(2.10)ẽ0=a0g1⋯gm,ẽi=ai∏j≠igj(X-1),1≤i≤m.
Let L be the extension field of 𝔽 with [L:𝔽]=d. Then the group ring
(2.11)F[X](Xp-1)=F〈σ〉≅F⊕L⊕⋯⊕L︸m
is a commutative semisimple 𝔽-algebra. Any code C≤𝔽n with an automorphism σ∈Mon(C) is a module for this algebra. Put ei:=ẽi(σ)∈𝔽[σ]. Then C=Ce0⊕Ce1⊕⋯⊕Cem with Ce0=C(σ), E=Ce1⊕⋯⊕Cem. Omitting the coordinates of E that correspond to the fixed points of σ, the codes Cei are L-linear codes of length t. Clearly dim𝔽(E)=d∑i=1mdimL(Cei). If C is self-dual then dim(E)=t(p-1)/2.


## 3. Extremal Ternary Codes of Length 48

LetC=C⊥≤𝔽348 be an extremal self-dual ternary code of length 48, so d(C)=15.


### 3.1. Large Primes

In this section we prove the main result of this paper.Theorem 3.1.
LetC=C⊥≤𝔽348 be an extremal self-dual code with an automorphism of prime order p≥5. Then C is one of the two known codes. So either C=Q48 is the extended quadratic residue code of length 48 with automorphism group
(3.1)Mon(Q48)=C2×PSL2(47)oforder25⋅3⋅23⋅47
or C=P48 is the Pless code with automorphism group
(3.2)Mon(P48)=C2×SL2(23)⋅2oforder26⋅3⋅11⋅23.Lemma 3.2.
Letσ∈Mon(C) be an automorphism of prime order p≥5. Then either p=47 and (t,f)=(1,1) or p=23 and (t,f)=(2,2) or p=11 and (t,f)=(4,4).Proof.
For the proof we use the notation of Lemma2.3. In particular we let K:=ker(πt)={(0,…,0,c1,…,cf)∈C(σ)} and put K*:={(c1,…,cf)∣(0,…,0,c1,…,cf)∈C(σ)}. Then
(3.3)K*≤F3f,d(K*)≥15,dim⁡(K*)≥f-t2.
Moreover tp+f=48.
(1) Ift=1, then p=47. If p=47, then t=f=1. So assume that p<47 and t=1. Then the code E has length p and dimension (p-1)/2, therefore p≥d(C)=15. So p≥17 and f≤48-17=31.
ThenK*≤𝔽3f has dimension (f-1)/2 and minimum distance d(K*)≥15. From the bounds given in [8] there is no such possibility for f≤31.
(2) Ift=2, then p=23. Assume that t=2. Since 2·p≤48 we get p≤23, and if p=23, then (t,f)=(2,2).
So assume thatp<23. The code E is a nonzero code of length 2p and minimum distance ≥15, so 2p≥15 and p is one of 11, 13, 17, 19, and f=26, 22, 14, 10. The code K*≤𝔽3f has dimension ≥f/2-1 and minimum distance ≥15. Again by [8] there is no such code.
(3)p≠13. For p=13 one now only has the possibility t=3 and f=9. The same argument as above constructs a code K*≤𝔽39 of dimension at least (f+t)/2-t=3 of minimum distance ≥15>f which is absurd.
(4) Ifp=11, then t=f=4. Otherwise t=3 and f=15 and the code K* as above has length 15, dimension ≥6, and minimum distance ≥15 which is impossible.
(5) Ifp=7, then t=f=6. Otherwise t=3,4,5 and f=27, 20, 13 and the code K* as above has dimension ≥(f+t)/2-t=12,8,4, length f, and minimum distance ≥15 which is impossible by [8].
(6)p≠7. Assume that p=7, then t=f=6 and the kernel K of the projection of C(σ) onto the first 42 components is trivial. So the image of the projection is 𝔽36⊗〈(1,1,1,1,1,1,1)〉; in particular it contains the vector (17,035) of weight 7. So C(σ) contains some word (17,035,a1,…,a6) of weight ≤13 which is a contradiction.
(7) Ifp=5, then t=f=8 or t=9andf=3. Otherwise t=3,4,5,6,7 and f=33,28,23,18,13 and the code K*≤𝔽3f has dimension ≥(f+t)/2-t=15,12,9,6,3 and minimum distance ≥15 which is impossible by [8].
(8)p≠5. Assume that p=5. Then one possibility is that t=8 and the projection of C(σ) onto the first 8·5 coordinates is 𝔽38⊗〈(1,1,1,1,1)〉 and contains a word of weight 5. But then C(σ) has a word of weight w with 5<w≤5+8=13 a contradiction.
The other possibility ist=9. Then the code E=E⊥ is a Hermitian self-dual code of length 9 over the field with 34=81 elements, which is impossible, since the length of such a code is 2 times the dimension and hence even.Lemma 3.3.
Ifp=11, then C≅P48.Proof.
Letσ∈Mon(C) be of order 11. Since (x11-1)=(x-1)gh∈𝔽3[x] for irreducible polynomials g,h of degree 5,
(3.4)F3〈σ〉≅F3⊕F35⊕F35.
Let e1,e2,e3∈𝔽3〈σ〉 denote the primitive idempotents. Then C=Ce1⊕Ce2⊕Ce3 with C(σ)=Ce1=Ce1⊥ of dimension 4 and Ce2=Ce3⊥≤(𝔽35⊕𝔽35)4. Clearly the projection of C(σ) onto the first 44 coordinates is injective. Since all weights of C are multiples of 3 and ≥15, this leaves just one possibility for C(σ):
(3.5)G0=(L0∣R0):=(111011011011111101111101101111-1-10110111110111-11-10110110111111-1-11).
The cyclic code Z of length 11 with generator polynomial (x-1)g (and similarly the one with generator polynomial (x-1)h) has weight enumerator
(3.6)x11+132x5y6+110x2y9.
In particular it contains more words of weight 6 than of weight 9. This shows that the dimension of Cei over 𝔽35 is 2 for both i=2,3, since otherwise one of them has dimension ≥3 and therefore contains all words (0,0,c,αc) for all c∈Z and some α∈𝔽35. Not all of them can have weight ≥15. Similarly one sees that the codes Cei≤𝔽354 have minimum distance 3 for i=2,3. So we may choose generator matrices
(3.7)G1:=(10ab01cd),G2:=(10a′b′01c′d′)
with (abcd)∈GL2(𝔽35) and (a′b′c′d′)=-(abcd)-tr. To obtain 𝔽3-generator matrices for the corresponding codes Ce2 and Ce3 of length 48, we choose a generator matrix g1∈𝔽35×11 of the cyclic code Z of length 11 with generator polynomial (x-1)g and the corresponding dual basis g2∈𝔽35×11 of the cyclic code with generator polynomial (x-1)h. We compute the action of σ (the multiplication with x) and represent this as left multiplication with z11∈𝔽35×5 on the basis g1. If a=∑i=04aiz11i∈𝔽35 with ai∈𝔽3, then the entry a in G1 is replaced by ∑i=04aiz11ig1∈𝔽35×11and analogously for G2, where we use of course the matrix g2 instead of g1. Replacing the code by an equivalent one we may choose a, b, c as orbit representatives of the action of 〈-z11〉 on 𝔽35*.
A generator matrix ofC is then given by(3.8)(L0R0G10G20).
All codes obtained this way are equivalent to the Pless code P48.Lemma 3.4.
Ifp=23, then C≅P48 or C≅Q48.Proof.
Letσ∈Mon(C) be of order 23. Since (x23-1)=(x-1)gh∈𝔽3[x] for irreducible polynomials g,h of degree 11,
(3.9)F3〈σ〉≅F3⊕F311⊕F311.
Let e1,e2,e3∈𝔽3〈σ〉 denote the primitive idempotents. Then C=Ce1⊕Ce2⊕Ce3 with C(σ)=Ce1=Ce1⊥ of dimension 2 and Ce2=Ce3⊥≤(𝔽311⊕𝔽311)2. Since all weights of C are multiples of 3, this leaves just one possibility for C(σ) (up to equivalence):
(3.10)C(σ)=〈(123,023,1,0),(023,123,0,1)〉.
The codes Ce2 and Ce3 are codes of length 2 over 𝔽311 such that dim(Ce2)+dim(Ce3)=2. Note that the alphabet 𝔽311 is identified with the cyclic code of length 23 with generator polynomial (x-1)g (resp., (x-1)h). These codes have minimum distance 9<15, so dim(Ce2)=dim(Ce3)=1 and both codes have a generator matrix of the form (1,t) (resp., (1,-t-1)) for t∈𝔽311*. Going through all possibilities for t (up to the action of the subgroup of 𝔽311* of order 23) the only codes C for which C(σ)⊕Ce2⊕Ce3 have minimum distance ≥15 are the two known extremal codes P48 and Q48.Lemma 3.5.
Ifp=47, then C≅Q48.Proof.
The subcodeC0:={c∈𝔽347|(c,0)∈C} is a cyclic code of length 47, dimension 23, and minimum distance ≥15. Since x47-1=(x-1)gh∈𝔽3[x] for irreducible polynomials g,h of degree 23, C0 is the cyclic code with generator polynomial (x-1)g (or equivalently (x-1)h) and C=〈(C0,0),1〉≤𝔽348 is the extended quadratic residue code.


### 3.2. Automorphisms of Order 2

As above letC=C⊥≤𝔽348 be an extremal self-dual ternary code. Assume that σ∈Mon(C) such that the permutational part π(σ) has order 2. Then σ2=±1 because of Lemma 2.1. If σ2=-1, then σ is conjugate to a block diagonal matrix with all blocks (01-10)=:J and C is a Hermitian self-dual code of length 24 over 𝔽9. Such automorphisms σ with σ2=-1 occur for both known extremal codes.Ifσ2=1, then σ is conjugate to a block diagonal matrix(3.11)σ~diag⁡((0110)t,1f,(-1)a)
for t,a,f∈ℕ0, 2t+a+f=48.Proposition 3.6.
Assume thatσ∈Mon(C), σ2=1 and π(σ)≠1. Then either (t,a,f)=(24,0,0) or (t,a,f)=(22,2,2). Automorphisms of both kinds are contained in Aut(P48).Proof.
(1) Wlogf≤a: Replacing σ by -σ we may assume without loss of generality that f≤a.
(2)f-t∈4ℤ: By Lemma 2.3 the code C(σ)*≤𝔽3t+f is a self-dual code with respect to the inner product (x,y)=-∑i=1txiyi+∑j=1fxjyj. This space only contains a self-dual code if f-t is a multiple of 4.
(3)t+f∈{22,24}: The code C(σ)*≤𝔽3t+f has dimension (t+f)/2 and minimum distance ≥15/2 and hence minimum distance ≥8. By [8] this implies that t+f≥22. Since t+a≥t+f and (t+a)+(t+f)=48 this only leaves these two possibilities.
(4)t+f≠22: We first treat the case f≤14. Then K*≅ker(πt) is a code of length f≤14 and minimum distance ≥15 and hence trivial. So πt is injective and
(3.12)C(σ)≅D:=πt(C(σ))≤F3t,dim⁡(D)=11,d(D)≥⌈15-f2⌉.
Using [8] and the fact that f-t is a multiple of 4, this only leaves the cases (t,f)∈{(19,3),(21,1)}. To rule out these two cases we use the fact that D is the dual of the self-orthogonal ternary code D⊥=πt(ker(πf)). The bounds in [9] give d(D)≤5<(15-3)/2 for t=19 and d(D)≤6<(15-1)/2 for t=21.
Iff≥15, then t≤7 and K*≅ker(πt) has dimension f-t>0 and minimum distance ≥15. This is easily ruled out by the known bounds (see [8]).
(5) Ift+f=24 then either (t,f)=(24,0) or (t,f)=(22,2). Again the case f>t is easily ruled out using dimension and minimum distance of K* as before.
So assume thatf≤t, and let D=πt(C(σ)) as before. Then dim(D)=12 and using [8] one gets that
(3.13)(t,f)∈{(24,0),(22,2),(20,4)}.
Assume that t=20. Then there is some self-dual code Λ=Λ⊥≤𝔽320 such that
(3.14)D⊥=πt(ker⁡(πf))≤Λ=Λ⊥≤D.
Clearly also d(Λ)≥d(D)≥6, so Λ is an extremal ternary code of length 20. There are 6 such codes, and none of them has a proper overcode with minimum distance 6.Remark 3.7.
Ifσ∈Mon(C) is some automorphism of order 4, then σ2=-1 or σ2 has type (24,0,0) in the notation of Proposition 3.6.Proof.
Assume thatσ∈Mon(C) has order 4 but σ2≠-1. Then τ=σ2 is one of the automorphisms from Proposition 3.6 and so σ is conjugate to some block diagonal matrix
(3.15)σ~diag⁡((0100001000011000)t/2,(0110)f/2,(0-110)a/2).
If t=22 and f=2 then the fixed code of σ is a self-dual code in 〈(1,1,1,1)〉t/2⦹〈(1,1)〉f/2 and C(σ)*≤𝔽3t/2+f/2 is a self-dual code with respect to the form (x,y):=∑i=1t/2xiyi-∑i=t/2+1t/2+f/2xiyi which implies that t/2-f/2 is a multiple of 4, a contradiction.For the two known extremal codes all automorphismsσ of order 4 satisfy σ2=-1. It would be nice to have some argument to exclude the other possibility.


## 3.1. Large Primes

In this section we prove the main result of this paper.Theorem 3.1.
LetC=C⊥≤𝔽348 be an extremal self-dual code with an automorphism of prime order p≥5. Then C is one of the two known codes. So either C=Q48 is the extended quadratic residue code of length 48 with automorphism group
(3.1)Mon(Q48)=C2×PSL2(47)oforder25⋅3⋅23⋅47
or C=P48 is the Pless code with automorphism group
(3.2)Mon(P48)=C2×SL2(23)⋅2oforder26⋅3⋅11⋅23.Lemma 3.2.
Letσ∈Mon(C) be an automorphism of prime order p≥5. Then either p=47 and (t,f)=(1,1) or p=23 and (t,f)=(2,2) or p=11 and (t,f)=(4,4).Proof.
For the proof we use the notation of Lemma2.3. In particular we let K:=ker(πt)={(0,…,0,c1,…,cf)∈C(σ)} and put K*:={(c1,…,cf)∣(0,…,0,c1,…,cf)∈C(σ)}. Then
(3.3)K*≤F3f,d(K*)≥15,dim⁡(K*)≥f-t2.
Moreover tp+f=48.
(1) Ift=1, then p=47. If p=47, then t=f=1. So assume that p<47 and t=1. Then the code E has length p and dimension (p-1)/2, therefore p≥d(C)=15. So p≥17 and f≤48-17=31.
ThenK*≤𝔽3f has dimension (f-1)/2 and minimum distance d(K*)≥15. From the bounds given in [8] there is no such possibility for f≤31.
(2) Ift=2, then p=23. Assume that t=2. Since 2·p≤48 we get p≤23, and if p=23, then (t,f)=(2,2).
So assume thatp<23. The code E is a nonzero code of length 2p and minimum distance ≥15, so 2p≥15 and p is one of 11, 13, 17, 19, and f=26, 22, 14, 10. The code K*≤𝔽3f has dimension ≥f/2-1 and minimum distance ≥15. Again by [8] there is no such code.
(3)p≠13. For p=13 one now only has the possibility t=3 and f=9. The same argument as above constructs a code K*≤𝔽39 of dimension at least (f+t)/2-t=3 of minimum distance ≥15>f which is absurd.
(4) Ifp=11, then t=f=4. Otherwise t=3 and f=15 and the code K* as above has length 15, dimension ≥6, and minimum distance ≥15 which is impossible.
(5) Ifp=7, then t=f=6. Otherwise t=3,4,5 and f=27, 20, 13 and the code K* as above has dimension ≥(f+t)/2-t=12,8,4, length f, and minimum distance ≥15 which is impossible by [8].
(6)p≠7. Assume that p=7, then t=f=6 and the kernel K of the projection of C(σ) onto the first 42 components is trivial. So the image of the projection is 𝔽36⊗〈(1,1,1,1,1,1,1)〉; in particular it contains the vector (17,035) of weight 7. So C(σ) contains some word (17,035,a1,…,a6) of weight ≤13 which is a contradiction.
(7) Ifp=5, then t=f=8 or t=9andf=3. Otherwise t=3,4,5,6,7 and f=33,28,23,18,13 and the code K*≤𝔽3f has dimension ≥(f+t)/2-t=15,12,9,6,3 and minimum distance ≥15 which is impossible by [8].
(8)p≠5. Assume that p=5. Then one possibility is that t=8 and the projection of C(σ) onto the first 8·5 coordinates is 𝔽38⊗〈(1,1,1,1,1)〉 and contains a word of weight 5. But then C(σ) has a word of weight w with 5<w≤5+8=13 a contradiction.
The other possibility ist=9. Then the code E=E⊥ is a Hermitian self-dual code of length 9 over the field with 34=81 elements, which is impossible, since the length of such a code is 2 times the dimension and hence even.Lemma 3.3.
Ifp=11, then C≅P48.Proof.
Letσ∈Mon(C) be of order 11. Since (x11-1)=(x-1)gh∈𝔽3[x] for irreducible polynomials g,h of degree 5,
(3.4)F3〈σ〉≅F3⊕F35⊕F35.
Let e1,e2,e3∈𝔽3〈σ〉 denote the primitive idempotents. Then C=Ce1⊕Ce2⊕Ce3 with C(σ)=Ce1=Ce1⊥ of dimension 4 and Ce2=Ce3⊥≤(𝔽35⊕𝔽35)4. Clearly the projection of C(σ) onto the first 44 coordinates is injective. Since all weights of C are multiples of 3 and ≥15, this leaves just one possibility for C(σ):
(3.5)G0=(L0∣R0):=(111011011011111101111101101111-1-10110111110111-11-10110110111111-1-11).
The cyclic code Z of length 11 with generator polynomial (x-1)g (and similarly the one with generator polynomial (x-1)h) has weight enumerator
(3.6)x11+132x5y6+110x2y9.
In particular it contains more words of weight 6 than of weight 9. This shows that the dimension of Cei over 𝔽35 is 2 for both i=2,3, since otherwise one of them has dimension ≥3 and therefore contains all words (0,0,c,αc) for all c∈Z and some α∈𝔽35. Not all of them can have weight ≥15. Similarly one sees that the codes Cei≤𝔽354 have minimum distance 3 for i=2,3. So we may choose generator matrices
(3.7)G1:=(10ab01cd),G2:=(10a′b′01c′d′)
with (abcd)∈GL2(𝔽35) and (a′b′c′d′)=-(abcd)-tr. To obtain 𝔽3-generator matrices for the corresponding codes Ce2 and Ce3 of length 48, we choose a generator matrix g1∈𝔽35×11 of the cyclic code Z of length 11 with generator polynomial (x-1)g and the corresponding dual basis g2∈𝔽35×11 of the cyclic code with generator polynomial (x-1)h. We compute the action of σ (the multiplication with x) and represent this as left multiplication with z11∈𝔽35×5 on the basis g1. If a=∑i=04aiz11i∈𝔽35 with ai∈𝔽3, then the entry a in G1 is replaced by ∑i=04aiz11ig1∈𝔽35×11and analogously for G2, where we use of course the matrix g2 instead of g1. Replacing the code by an equivalent one we may choose a, b, c as orbit representatives of the action of 〈-z11〉 on 𝔽35*.
A generator matrix ofC is then given by(3.8)(L0R0G10G20).
All codes obtained this way are equivalent to the Pless code P48.Lemma 3.4.
Ifp=23, then C≅P48 or C≅Q48.Proof.
Letσ∈Mon(C) be of order 23. Since (x23-1)=(x-1)gh∈𝔽3[x] for irreducible polynomials g,h of degree 11,
(3.9)F3〈σ〉≅F3⊕F311⊕F311.
Let e1,e2,e3∈𝔽3〈σ〉 denote the primitive idempotents. Then C=Ce1⊕Ce2⊕Ce3 with C(σ)=Ce1=Ce1⊥ of dimension 2 and Ce2=Ce3⊥≤(𝔽311⊕𝔽311)2. Since all weights of C are multiples of 3, this leaves just one possibility for C(σ) (up to equivalence):
(3.10)C(σ)=〈(123,023,1,0),(023,123,0,1)〉.
The codes Ce2 and Ce3 are codes of length 2 over 𝔽311 such that dim(Ce2)+dim(Ce3)=2. Note that the alphabet 𝔽311 is identified with the cyclic code of length 23 with generator polynomial (x-1)g (resp., (x-1)h). These codes have minimum distance 9<15, so dim(Ce2)=dim(Ce3)=1 and both codes have a generator matrix of the form (1,t) (resp., (1,-t-1)) for t∈𝔽311*. Going through all possibilities for t (up to the action of the subgroup of 𝔽311* of order 23) the only codes C for which C(σ)⊕Ce2⊕Ce3 have minimum distance ≥15 are the two known extremal codes P48 and Q48.Lemma 3.5.
Ifp=47, then C≅Q48.Proof.
The subcodeC0:={c∈𝔽347|(c,0)∈C} is a cyclic code of length 47, dimension 23, and minimum distance ≥15. Since x47-1=(x-1)gh∈𝔽3[x] for irreducible polynomials g,h of degree 23, C0 is the cyclic code with generator polynomial (x-1)g (or equivalently (x-1)h) and C=〈(C0,0),1〉≤𝔽348 is the extended quadratic residue code.


## 3.2. Automorphisms of Order 2

As above letC=C⊥≤𝔽348 be an extremal self-dual ternary code. Assume that σ∈Mon(C) such that the permutational part π(σ) has order 2. Then σ2=±1 because of Lemma 2.1. If σ2=-1, then σ is conjugate to a block diagonal matrix with all blocks (01-10)=:J and C is a Hermitian self-dual code of length 24 over 𝔽9. Such automorphisms σ with σ2=-1 occur for both known extremal codes.Ifσ2=1, then σ is conjugate to a block diagonal matrix(3.11)σ~diag⁡((0110)t,1f,(-1)a)
for t,a,f∈ℕ0, 2t+a+f=48.Proposition 3.6.
Assume thatσ∈Mon(C), σ2=1 and π(σ)≠1. Then either (t,a,f)=(24,0,0) or (t,a,f)=(22,2,2). Automorphisms of both kinds are contained in Aut(P48).Proof.
(1) Wlogf≤a: Replacing σ by -σ we may assume without loss of generality that f≤a.
(2)f-t∈4ℤ: By Lemma 2.3 the code C(σ)*≤𝔽3t+f is a self-dual code with respect to the inner product (x,y)=-∑i=1txiyi+∑j=1fxjyj. This space only contains a self-dual code if f-t is a multiple of 4.
(3)t+f∈{22,24}: The code C(σ)*≤𝔽3t+f has dimension (t+f)/2 and minimum distance ≥15/2 and hence minimum distance ≥8. By [8] this implies that t+f≥22. Since t+a≥t+f and (t+a)+(t+f)=48 this only leaves these two possibilities.
(4)t+f≠22: We first treat the case f≤14. Then K*≅ker(πt) is a code of length f≤14 and minimum distance ≥15 and hence trivial. So πt is injective and
(3.12)C(σ)≅D:=πt(C(σ))≤F3t,dim⁡(D)=11,d(D)≥⌈15-f2⌉.
Using [8] and the fact that f-t is a multiple of 4, this only leaves the cases (t,f)∈{(19,3),(21,1)}. To rule out these two cases we use the fact that D is the dual of the self-orthogonal ternary code D⊥=πt(ker(πf)). The bounds in [9] give d(D)≤5<(15-3)/2 for t=19 and d(D)≤6<(15-1)/2 for t=21.
Iff≥15, then t≤7 and K*≅ker(πt) has dimension f-t>0 and minimum distance ≥15. This is easily ruled out by the known bounds (see [8]).
(5) Ift+f=24 then either (t,f)=(24,0) or (t,f)=(22,2). Again the case f>t is easily ruled out using dimension and minimum distance of K* as before.
So assume thatf≤t, and let D=πt(C(σ)) as before. Then dim(D)=12 and using [8] one gets that
(3.13)(t,f)∈{(24,0),(22,2),(20,4)}.
Assume that t=20. Then there is some self-dual code Λ=Λ⊥≤𝔽320 such that
(3.14)D⊥=πt(ker⁡(πf))≤Λ=Λ⊥≤D.
Clearly also d(Λ)≥d(D)≥6, so Λ is an extremal ternary code of length 20. There are 6 such codes, and none of them has a proper overcode with minimum distance 6.Remark 3.7.
Ifσ∈Mon(C) is some automorphism of order 4, then σ2=-1 or σ2 has type (24,0,0) in the notation of Proposition 3.6.Proof.
Assume thatσ∈Mon(C) has order 4 but σ2≠-1. Then τ=σ2 is one of the automorphisms from Proposition 3.6 and so σ is conjugate to some block diagonal matrix
(3.15)σ~diag⁡((0100001000011000)t/2,(0110)f/2,(0-110)a/2).
If t=22 and f=2 then the fixed code of σ is a self-dual code in 〈(1,1,1,1)〉t/2⦹〈(1,1)〉f/2 and C(σ)*≤𝔽3t/2+f/2 is a self-dual code with respect to the form (x,y):=∑i=1t/2xiyi-∑i=t/2+1t/2+f/2xiyi which implies that t/2-f/2 is a multiple of 4, a contradiction.For the two known extremal codes all automorphismsσ of order 4 satisfy σ2=-1. It would be nice to have some argument to exclude the other possibility.



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*Source: 154281-2012-01-23.xml*