Lattices from codes over $\mathbb{Z}_q$: Generalization of Constructions $D$, $D'$ and $\overline{D}$

TL;DR

This paper generalizes classical lattice constructions from codes over finite fields to codes over integer rings, introducing zero-one addition and establishing conditions for lattice formation, with implications for cryptography.

Contribution

It extends lattice constructions D, D', and Forney's formula to codes over z_q, defining zero-one addition and linking it to lattice properties, also generalizing Construction A' to the real case.

Findings

Construction z_q produces lattices when codes are closed under zero-one addition.

Extended Construction z_q is equivalent to lattice formation under specific closure conditions.

Generalization of Construction A' to real case involves shifted zero-one addition for lattice criteria.

Abstract

In this paper, we extend the lattice Constructions $D$, $D'$ and $\overline{D}$ $($this latter is also known as Forney's code formula$)$ from codes over $\mathbb{F}_p$ to linear codes over $\mathbb{Z}_q$, where $q \in \mathbb{N}$. We define an operation in $\mathbb{Z}_q^n$ called zero-one addition, which coincides with the Schur product when restricted to $\mathbb{Z}_2^n$ and show that the extended Construction $\overline{D}$ produces a lattice if and only if the nested codes are closed under this addition. A generalization to the real case of the recently developed Construction $A'$ is also derived and we show that this construction produces a lattice if and only if the corresponding code over $\mathbb{Z}_q[X]/X^a$ is closed under a shifted zero-one addition. One of the motivations for this work is the recent use of $q$-ary lattices in cryptography.