Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex