On extremal self-dual ternary codes of length 48

All extremal ternary codes of length 48 that have some automorphism of prime order $p\geq 5$ are equivalent to one of the two known codes, the Pless code or the extended quadratic residue code.


Introduction.
The notion of an extremal code has been introduced in [8]. As Andrew Gleason [4] 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 [6] and [3] for the first papers on this subject.
Ternary codes have been studied in [7]. The minimum distance d(C) := min{wt(c) | 0 = c ∈ C} of a self-dual ternary code C = C ⊥ ≤ F n 3 of length n is bounded by 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 Q 24 :=QR(23, 3) and the Pless code P 24 ). For length 36, the Pless code yields one example of an extremal code. [7] 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 Q 48 and the Pless code P 48 . 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.
Let F be some finite field, F * its multiplicative group. For any monomial transformation σ ∈ Mon n (F) := F * ≀ S n , the image π(σ) ∈ S n is called the permutational part of σ. Then σ has a unique expression as and m(σ) is called the monomial part of σ. For a code C ≤ F n we let be the full monomial automorphism group of C. We call a code C ≤ F n an orthogonal direct sum, if there are codes Lemma 2.1. Let C ≤ F n be not an orthogonal direct sum. Then the kernel of the restriction of π to Mon(C) is isomorphic to F * .
In the investigation of possible automorphisms of codes, the following strategy has proved to be very fruitful ( [6], [2]). Definition 2.2. Let σ ∈ Mon(C) be an automorphism of C. Then π(σ) ∈ S n 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. Let σ ∈ Mon(C) have prime order p. (a) If p does not divide |F * | then there is some element τ ∈ Mon n (F) such that m(τ στ −1 ) = id. Replacing C by τ (C) we hence may assume that m(σ) = 1.
The way to analyse the code E from Lemma 2.3 is based on the following remark.
Remark 2.5. Let p = char(F) be some prime and σ ∈ Mon n (F) be an element of order p. Let be the factorization of X p − 1 into irreducible polynomials. Then all factors g i have the same Then the primitive idempotents in F[X]/(X p − 1) are given by the classes of Let L be the extension field of F with [L : F] = d. Then the group ring Omitting the coordinates of E that correspond to the fixed points of σ, the codes Ce i are L-linear codes of length t.
Let C = C ⊥ ≤ F 48 3 be an extremal self-dual ternary code of length 48, so d(C) = 15.

Large primes.
In this section we prove the main result of this paper.
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 = Q 48 is the extended quadratic residue code of length 48 with automorphism group Mon(C) = C 2 × PSL 2 (47) of order 2 5 · 3 · 23 · 47 or C = P 48 is the Pless code with automorphism group Mon(C) = C 2 × SL 2 (23).2 of order 2 6 · 3 · 11 · 23. Proof. For the proof we use the notation of Lemma 2.3. In particular we let K := ker Moreover tp + f = 48. Then K * ≤ F f 3 has dimension (f − 1)/2 and minimum distance d(K * ) ≥ 15. From the bounds given in [5] there is no such possibility for f ≤ 31.
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 * ≤ F 9 3 of dimension at least (f + t)/2 − t = 3 of minimum distance ≥ 15 > f which is absurd. 4) If p = 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) If p = 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 , minimum distance ≥ 15 which is impossible by [5]. 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 F 6 3 ⊗ (1, 1, 1, 1, 1, 1, 1) , in particular it contains the vector (1 7 , 0 35 ) of weight 7. So C(σ) contains some word (1 7 , 0 35 , a 1 , . . . , a 6 ) of weight ≤ 13 which is a contradiction. 7) If p = 5 then t = f = 8 or t = 9 and f = 3. Otherwise t = 3, 4, 5, 6, 7 and f = 33, 28, 23, 18, 13 and the code K * ≤ F f 3 has dimension ≥ (f + t)/2 − t = 15, 12, 9, 6, 3 and minimum distance ≥ 15 which is impossible by [5]. 8) p = 5. Assume that p = 5. Then either t = 8 and the projection of C(σ) onto the first 8·5 coordinates is F 8 3 ⊗ (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 is t = 9. Then the code E = E ⊥ is a Hermitian self-dual code of length 9 over the field with 3 4 = 81 elements, which is impossible, since the length of such a code is 2 times the dimension and hence even. Proof. Let σ ∈ Mon(C) be of order 11. Since (x 11 − 1) = (x − 1)gh ∈ F 3 [x] for irreducible polynomials g, h of degree 5, Let e 1 , e 2 , e 3 ∈ F 3 σ denote the primitive idempotents. Then C = Ce 1 ⊕ Ce 2 ⊕ Ce 3 with C(σ) = Ce 1 = Ce ⊥ 1 of dimension 4 and Ce 2 = Ce ⊥ 3 ≤ (F 3 5 ⊕ F 3 5 ) 4 . Clearly the projection of C(σ) onto the first 44 coordinates is injective. Since all weights of C are multiples of 3 and 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 x 11 + 132x 5 y 6 + 110x 2 y 9 in particular it contains more words of weight 6 than of weight 9. This shows that the dimension of Ce i over F 3 5 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 ∈ F 3 5 . Not all of them can have weight ≥ 15. Similarly one sees that the codes Ce i ≤ F 4 3 5 have minimum distance 3 for i = 2, 3. So we may choose generator matrices . To obtain F 3 -generator matrices for the corresponding codes Ce 2 and Ce 3 of length 48, we choose a generator matrix g 1 ∈ F 5×11 3 of the cyclic code Z of length 11 with generator polynomial (x − 1)g, and the corresponding dual basis g 2 ∈ F 5×11 3 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 z 11 ∈ F 5×5 3 on the basis g 1 . If a = 4 i=0 a i z i 11 ∈ F 3 5 with a i ∈ F 3 , then the entry a in G1 is replaced by . Analogously for G2, where we use of course the matrix g 2 instead of g 1 . Replacing the code by an equivalent one we may choose a, b, c as orbit representatives of the action of −z 11 on F * 3 5 . A generator matrix of C is then given by All codes obtained this way are equivalent to the Pless code P 48 . Proof. Let σ ∈ Mon(C) be of order 23. Since (x 23 − 1) = (x − 1)gh ∈ F 3 [x] for irreducible polynomials g, h of degree 11, Let e 1 , e 2 , e 3 ∈ F 3 σ denote the primitive idempotents. Then C = Ce 1 ⊕ Ce 2 ⊕ Ce 3 with C(σ) = Ce 1 = Ce ⊥ 1 of dimension 2 and Ce 2 = Ce ⊥ 3 ≤ (F 3 11 ⊕ F 3 11 ) 2 . Since all weights of C are multiples of 3, this leaves just one possibility for C(σ) (up to equivalence): (1 23 , 0 23 , 1, 0), (0 23 , 1 23 , 0, 1) .
The codes Ce 2 and Ce 3 are codes of length 2 over F 3 11 such that dim(Ce 2 ) + dim(Ce 3 ) = 2. Note that the alphabet F 3 11 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(Ce 2 ) = dim(Ce 3 ) = 1 and both codes have a generator matrix of the form (1, t) (resp. (1, −t −1 )) for t ∈ F * 3 11 . Going through all possibilities for t (up to the action of the subgroup of F * 3 11 of order 23) the only codes C for which C(σ) ⊕ Ce 2 ⊕ Ce 3 have minimum distance ≥ 15 are the two known extremal codes P 48 and Q 48 . Proof. The subcode C 0 := {c ∈ F 47 3 | (c, 0) ∈ C} is a cyclic code of length 47, dimension 23 and minimum distance ≥ 15. Since x 47 − 1 = (x − 1)gh ∈ F 3 [x] for irreducible polynomials g, h of degree 23, C 0 is the cyclic code with generator polynomial (x − 1)g (or equivalently (x − 1)h) and C = (C 0 , 0), 1 ≤ F 48 3 is the extended quadratic residue code.
As above let C = C ⊥ ≤ F 48 3 be an extremal self-dual ternary code. Assume that σ ∈ Mon(C) such that the permutational part π(σ) has order 2. Then σ 2 = ±1 because of Lemma 2.1. If σ 2 = −1, then σ is conjugate to a block diagonal matrix with all blocks 0 1 −1 0 =: J and C is a Hermitian self-dual code of length 24 over F 9 . Such automorphisms σ with σ 2 = −1 occur for both known extremal codes. If σ 2 = 1, then σ is conjugate to a block diagonal matrix for t, a, f ∈ N 0 , 2t + a + f = 48.
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) * ≤ F t+f 3 is a self-dual code with respect to the inner product (x, y) = − t i=1 x i y i + f j=1 x j y j . This space only contains a self-dual code if f −t is a multiple of 4.
Clearly also d(Λ) ≥ d(D) ≥ 6, so Λ is an extremal ternary code of length 20. There are 6 such codes, none of them has a proper overcode with minimum distance 6.
Proof. Assume that σ ∈ Mon(C) has order 4 but σ 2 = −1. Then τ = σ 2 is one of the automorphisms from Proposition 3.6 and so σ is conjugate to some block diagonal matrix σ ∼ diag(