A note on unimodular lattices with trivial automorphism groups

Abstract

In this paper, we give a simple construction method of positive definite unimodular lattices with trivial automorphism groups (i.e., the tri-diagonal matrix method), and obtain odd positive definite unimodular lattices of dimensions 29≤n≤42 and even positive definite unimodular lattices of dimensions n=32, 40 with trivial automorphism groups. Combining these with Bannai’s and Bacher and Venkov’s results we obtain that there exists a unimodular lattice with a trivial automorphism group if and only if the dimension n>28.