Abstract
In this paper we construct new extremal and optimal unimodular lattices in dimensions 36, 38, 42, 45, 52, 54, 60 and 68. We construct them in two ways: first in the case of dimensions congruent to 4 modulo 8 by construction B 3 followed by density doubling, generalizing the constructions of Sphere Packing Lattices and Groups (1988) 148 and Théorie des Nombres (1989) 772; and second by applying the well known Construction A to self-dual codes over GF(5) and to codes over the ring Z/25 Z. In particular the lattice in dimension 60, P 60 q , generalizes the construction of the lattice P 48 q . We also give the complete weight enumerator of the extended ternary quadratic residue code of length 60 and we provide a table of the best known unimodular lattices of dimensions up to 80.
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