Abstract
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.
Highlights
The notion of an extremal self-dual code has been introduced in 1
The present paper investigates the extremal codes of length 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
Summary
The notion of an extremal self-dual code has been introduced in 1. The most wanted extremal code is a binary self-dual doubly even code of length 72 and minimum distance 16. The minimum distance d C : min{wt c | 0 / c ∈ C} of a self-dual ternary code C C⊥ ≤ Fn3 of length n is bounded by dC. The present paper investigates the extremal codes of length 48. There are two such codes known, the extended quadratic. 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. 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
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