Modular Lattices from a Variation of Construction A over Number Fields

TL;DR

This paper introduces a new lattice construction method using number fields, providing explicit matrices and analyzing properties relevant to coding theory, including minimal norm and secrecy gain.

Contribution

It presents a generic construction of modular lattices from number fields with explicit matrices, focusing on properties important for coding and cryptography.

Findings

Explicit generator and Gram matrices for the lattices

Analysis of minimal norm, theta series, and kissing number

Identification of interesting lattices with potential coding applications

Abstract

We consider a variation of Construction A of lattices from linear codes based on two classes of number fields, totally real and CM Galois number fields. We propose a generic construction with explicit generator and Gram matrices, then focus on modular and unimodular lattices, obtained in the particular cases of totally real, respectively, imaginary, quadratic fields. Our motivation comes from coding theory, thus some relevant properties of modular lattices, such as minimal norm, theta series, kissing number and secrecy gain are analyzed. Interesting lattices are exhibited.