Bounds for the $l_1$-distance of $q$-ary lattices obtained via Constructions D, D$^{'}$ and $\overline{D}$

TL;DR

This paper investigates bounds on the minimum $l_1$-distance of $q$-ary lattices constructed via Constructions D, D', and 0D, establishing connections, explicit formulas, and conditions for generator matrices.

Contribution

It introduces bounds and explicit expressions for the minimum $l_1$-distances of lattices from Constructions D, D', and 0D, and explores their interrelations and conditions for generator matrices.

Findings

Bounds for the minimum $l_1$-distance of lattices $\\Lambda_{D}$, $\\Lambda_{D'}$, and $\\Lambda_{\ar{D}}$ are established.

Connections between Constructions D and D' are demonstrated.

Explicit formulas for minimum $l_1$-distances are derived under certain code chain conditions.

Abstract

Lattices have been used in several problems in coding theory and cryptography. In this paper we approach $q$-ary lattices obtained via Constructions D, $\D'$ and $\overline{D}$. It is shown connections between Constructions D and $\D'$. Bounds for the minimum $l_1$-distance of lattices $\Lambda_{D}$, $\Lambda_{D'}$ and $\Lambda_{\overline{D}}$ and, under certain conditions, a generator matrix for $\Lambda_{D'}$ are presented. In addition, when the chain of codes used is closed under the zero-one addition, we derive explicit expressions for the minimum $l_1$-distances of the lattices $\Lambda_{D}$ and $\Lambda_{\overline{D}}$ attached to the distances of the codes used in these constructions.