On some self-dual codes and unimodular lattices in dimension 48
The shadow of a singly-even self-dual code is the coset of the code in the orthogonal of the doubly-even subcode. This subcode consists of two cosets of the doubly-even subcode. In certain cases, the union of one of these cosets and the subcode of doubly-even vectors can form a doubly-even self-dual code. The authors use this to show that one of the two doubly even self-dual neighbors of an extremal doubly even self-dual \([48,24,10]\) code whose shadow has minimum weight 4 is an extremal doubly even self-dual \([48,24,12]\) code. Similarly for lattices they show that every extremal odd unimodular lattice in dimension 48 whose shadow has minimum norm 2 has an extremal even unimodular neighbor. Additionally extremal singly even self-dual \([48,24,10]\) codes with shadows whose minimum weight is 8 and extremal odd unimodular lattices in dimension 48 whose shadows have minimum norm 4 are studied.
scientific article; zbMATH DE number 2165399
scientific article