A ternary construction of lattices

TL;DR

This paper introduces a new ternary lattice construction method from three rows and codes, enabling the recovery of many known lattices and the creation of new extremal lattices in low dimensions.

Contribution

It presents a novel ternary construction approach that unifies existing lattices and facilitates the generation of new extremal lattices in dimensions 32, 40, and 48.

Findings

Most laminated and Kappa lattices in dimensions up to 24 are recoverable.

New extremal even lattices are constructed in dimensions 32, 40, and 48.

The ternary construction can generate many low-dimensional sub-optimal lattices.

Abstract

In this paper we propose a general ternary construction of lattices from three rows and ternary codes. Most laminated lattices and Kappa lattices in ${\bf R}^n$, $n\leq 24$ can be recovered from our tenary construction naturally. This ternary construction of lattices can be used to generate many new "sub-optimal" lattices of low dimensions.Based on this ternary construction new extremal even lattices of dimensions $32, 40$ and $48$ are also constructed.