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. 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 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 [5] 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 ([6]). 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