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

Abstract

In this paper, we extend the lattice Constructions $D$, $D^{\prime }$ 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 $A^{\prime }$ 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.

18 pages

References in corpus (1)

• Construction ${\pi }_A$ and ${\pi }_D$ Lattices: Construction, Goodness, and Decoding Algorithms