Construction of new extremal unimodular lattices

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 B3 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/25Z. In particular the lattice in dimension 60, P60q, generalizes the construction of the lattice P48q. 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.