Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80

TL;DR

This paper proves that extremal Type II lattices of ranks 40 and 80 are generated solely by vectors of norm min(L)+2, refining previous results that included vectors of norm min(L).

Contribution

It establishes that such lattices are generated only by vectors of norm min(L)+2, sharpening earlier results that included vectors of norm min(L) and min(L)+2.

Findings

Lattices of rank 40 and 80 are generated by vectors of norm min(L)+2.

This result refines previous findings by excluding vectors of norm min(L).

The proof enhances understanding of the structure of extremal Type II lattices.

Abstract

We show that, if L is an extremal Type II lattice of rank 40 or 80, then L is generated by its vectors of norm min(L)+2. This sharpens earlier results of Ozeki, and the second author and Abel, which showed that such lattices L are generated by their vectors of norms min(L) and min(L)+2.