On Extremal Self-Dual Ternary Codes of Length 48
All extremal ternary self-dual codes of length 48 that have some automorphism of prime order are equivalent to one of the two known codes, the Pless code or the extended quadratic residue code.
The notion of an extremal self-dual code has been introduced in . As Gleason  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 . The minimum distance of a self-dual ternary code of length 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 and the Pless code ). For length 36, the Pless code yields one example of an extremal code. Reference  shows that this is the only code with an automorphism of prime order ; 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 and the Pless code . The computer calculations described in this paper show that these two codes are the only extremal ternary codes of length 48 for which the order of the automorphism group is divisible by some prime . 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 (). A long-term project to find or even classify such lattices was my main motivation for this paper.
2. Automorphisms of Codes
Let be some finite field, its multiplicative group. For any monomial transformation , the image is called the permutational part of . Then has a unique expression as and is called the monomial part of . For a code we let be the full monomial automorphism group of .
We call a code an orthogonal direct sum, if there are codes () of length such that
Lemma 2.1. Let not be an orthogonal direct sum. Then the kernel of the restriction of to is isomorphic to .
Proof. Clearly since is an -subspace. Assume that with , not all equal. Let with pairwise distinct . Then is the direct sum of eigenspaces of . Moreover the standard basis is a basis of eigenvectors of so this is an orthogonal direct sum.
Definition 2.2. Let be an automorphism of . Then is a direct product of disjoint cycles of lengths dividing the order of . In particular if the order of is some prime , then we say that has cycle type , if has cycles of length and fixed points (so ).
Lemma 2.3. Let have prime order .(a)If does not divide then there is some element such that . Replacing by one hence may assume that .(b)Assume that does not divide , , and . Then , where is the fixed code of and is the unique -invariant complement of in .(c)Define two projections So . If is self-dual with respect to , then is a self-dual code with respect to the inner product .(d)In particular and .
In the following we will keep the notation of the previous lemma and regard the fixed code .
Remark 2.4. If then .
Proof. Otherwise the kernel is a nontrivial subcode of minimum distance .
The way to analyse the code from Lemma 2.3 is based on the following remark.
Remark 2.5. Let be some prime and an element of order . Let be the factorization of into irreducible polynomials. Then all factors have the same degree , the order of mod . There are polynomials () such that Then the primitive idempotents in are given by the classes of Let be the extension field of with . Then the group ring is a commutative semisimple -algebra. Any code with an automorphism is a module for this algebra. Put . Then with , . Omitting the coordinates of that correspond to the fixed points of , the codes are -linear codes of length . Clearly . If is self-dual then .
3. Extremal Ternary Codes of Length 48
Let be an extremal self-dual ternary code of length 48, so .
3.1. Large Primes
In this section we prove the main result of this paper.
Theorem 3.1. Let be an extremal self-dual code with an automorphism of prime order . Then is one of the two known codes. So either is the extended quadratic residue code of length 48 with automorphism group or is the Pless code with automorphism group
Lemma 3.2. Let be an automorphism of prime order . Then either and or and or and .
Proof. For the proof we use the notation of Lemma 2.3. In particular we let and put . Then
(1) If , then . If , then . So assume that and . Then the code has length and dimension , therefore . So and .
Then has dimension and minimum distance . From the bounds given in  there is no such possibility for .
(2) If , then . Assume that . Since we get , and if , then .
So assume that . The code is a nonzero code of length and minimum distance , so and is one of 11, 13, 17, 19, and , 22, 14, 10. The code has dimension and minimum distance . Again by  there is no such code.
(3) . For one now only has the possibility and . The same argument as above constructs a code of dimension at least of minimum distance which is absurd.
(4) If , then . Otherwise and and the code as above has length 15, dimension , and minimum distance which is impossible.
(5) If , then . Otherwise and , 20, 13 and the code as above has dimension , length , and minimum distance which is impossible by .
(6) . Assume that , then and the kernel of the projection of onto the first 42 components is trivial. So the image of the projection is ; in particular it contains the vector of weight 7. So contains some word of weight which is a contradiction.
(7) If , then or . Otherwise and and the code has dimension and minimum distance which is impossible by .
(8) . Assume that . Then one possibility is that and the projection of onto the first coordinates is and contains a word of weight 5. But then has a word of weight with a contradiction.
The other possibility is . Then the code is a Hermitian self-dual code of length 9 over the field with elements, which is impossible, since the length of such a code is 2 times the dimension and hence even.
Lemma 3.3. If , then .
Proof. Let be of order 11. Since for irreducible polynomials of degree 5,
Let denote the primitive idempotents. Then with of dimension 4 and . Clearly the projection of onto the first 44 coordinates is injective. Since all weights of are multiples of 3 and , this leaves just one possibility for :
The cyclic code of length 11 with generator polynomial (and similarly the one with generator polynomial ) has weight enumerator
In particular it contains more words of weight 6 than of weight 9. This shows that the dimension of over is 2 for both , since otherwise one of them has dimension and therefore contains all words for all and some . Not all of them can have weight . Similarly one sees that the codes have minimum distance 3 for . So we may choose generator matrices
with and . To obtain -generator matrices for the corresponding codes and of length 48, we choose a generator matrix of the cyclic code of length 11 with generator polynomial and the corresponding dual basis of the cyclic code with generator polynomial . We compute the action of (the multiplication with ) and represent this as left multiplication with on the basis . If with , then the entry in is replaced by and analogously for , where we use of course the matrix instead of . Replacing the code by an equivalent one we may choose , , as orbit representatives of the action of on .
A generator matrix of is then given by All codes obtained this way are equivalent to the Pless code .
Lemma 3.4. If , then or .
Proof. Let be of order 23. Since for irreducible polynomials of degree 11, Let denote the primitive idempotents. Then with of dimension 2 and . Since all weights of are multiples of 3, this leaves just one possibility for (up to equivalence): The codes and are codes of length 2 over such that . Note that the alphabet is identified with the cyclic code of length 23 with generator polynomial (resp., ). These codes have minimum distance , so and both codes have a generator matrix of the form (resp., ) for . Going through all possibilities for (up to the action of the subgroup of of order 23) the only codes for which have minimum distance are the two known extremal codes and .
Lemma 3.5. If , then .
Proof. The subcode is a cyclic code of length 47, dimension 23, and minimum distance . Since for irreducible polynomials of degree 23, is the cyclic code with generator polynomial (or equivalently ) and is the extended quadratic residue code.
3.2. Automorphisms of Order 2
As above let be an extremal self-dual ternary code. Assume that such that the permutational part has order 2. Then because of Lemma 2.1. If , then is conjugate to a block diagonal matrix with all blocks and is a Hermitian self-dual code of length 24 over . Such automorphisms with occur for both known extremal codes.
If , then is conjugate to a block diagonal matrix for , .
Proposition 3.6. Assume that , and . Then either or . Automorphisms of both kinds are contained in .
Proof. (1) Wlog : Replacing by we may assume without loss of generality that .
(2) : By Lemma 2.3 the code is a self-dual code with respect to the inner product . This space only contains a self-dual code if is a multiple of 4.
(3) : The code has dimension and minimum distance and hence minimum distance . By  this implies that . Since and this only leaves these two possibilities.
(4) : We first treat the case . Then is a code of length and minimum distance and hence trivial. So is injective and Using  and the fact that is a multiple of 4, this only leaves the cases . To rule out these two cases we use the fact that is the dual of the self-orthogonal ternary code . The bounds in  give for and for .
If , then and has dimension and minimum distance . This is easily ruled out by the known bounds (see ).
(5) If then either or . Again the case is easily ruled out using dimension and minimum distance of as before.
So assume that , and let as before. Then and using  one gets that Assume that . Then there is some self-dual code such that Clearly also , so 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. If is some automorphism of order 4, then or has type in the notation of Proposition 3.6.
Proof. Assume that has order 4 but . Then is one of the automorphisms from Proposition 3.6 and so is conjugate to some block diagonal matrix If and then the fixed code of is a self-dual code in and is a self-dual code with respect to the form which implies that is a multiple of 4, a contradiction.
For the two known extremal codes all automorphisms of order 4 satisfy . It would be nice to have some argument to exclude the other possibility.