arXiv subjects

Eleonesio Strey

Publications and source records attributed to Eleonesio Strey.

3 recordsLinked to original sources

arXiv2023

On Lattice Constructions D and D' from q-ary Linear Codes

Multilevel lattice codes, such as those associated to Constructions $C$, $\overline{D}$, D and D', have relevant applications in communications. In this paper, we investigate some properties of lattices obtained via Constructions D and D' from $q$-ary linear codes. Connections with Construction A, generator matrices, expressions and bounds for the lattice volume and minimum distances are derived. Extensions of previous results regarding construction and decoding of binary and $p$-ary linear codes ($p$ prime) are also presented.

cs.IT↗

arXiv2016

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

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 $Λ_{D}$, $Λ_{D'}$ and $Λ_{\overline{D}}$ and, under certain conditions, a generator matrix for $Λ_{D'}$ 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 $Λ_{D}$ and $Λ_{\overline{D}}$ attached to the distances of the codes used in these constructions.

cs.IT↗

arXiv2016

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

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.

cs.IT↗