Connections Between Construction D and Related Constructions of Lattices

Most practical constructions of lattice codes with high coding gains are multilevel constructions where each level corresponds to an underlying code component. Construction D, Construction D$'$, and Forney's code formula are classical constructions that produce such lattices explicitly from a family of nested binary linear codes. In this paper, we investigate these three closely related constructions along with the recently developed Construction A$'$ of lattices from codes over the polynomial ring $\mathbb{F}_2[u]/u^a$. We show that Construction by Code Formula produces a lattice packing if and only if the nested codes being used are closed under Schur product, thus proving the similarity of Construction D and Construction by Code Formula when applied to Reed-Muller codes. In addition, we relate Construction by Code Formula to Construction A$'$ by finding a correspondence between nested binary codes and codes over $\mathbb{F}_2[u]/u^a$. This proves that any lattice constructible using Construction by Code Formula is also constructible using Construction A$'$. Finally, we show that Construction A$'$ produces a lattice if and only if the corresponding code over $\mathbb{F}_2[u]/u^a$ is closed under shifted Schur product.Comment: Submitted to Designs, Codes and Cryptograph

Lattices from codes over Zq\mathbb{Z}_q: Generalization of Constructions DD, D′D' and D‾\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.Comment: 18 page

Construction πA\pi_A and πD\pi_D Lattices: Construction, Goodness, and Decoding Algorithms

A novel construction of lattices is proposed. This construction can be thought of as a special class of Construction A from codes over finite rings that can be represented as the Cartesian product of $L$ linear codes over $\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$, respectively, and hence is referred to as Construction $\pi_A$. The existence of a sequence of such lattices that is good for channel coding (i.e., Poltyrev-limit achieving) under multistage decoding is shown. A new family of multilevel nested lattice codes based on Construction $\pi_A$ lattices is proposed and its achievable rate for the additive white Gaussian channel is analyzed. A generalization named Construction $\pi_D$ is also investigated which subsumes Construction A with codes over prime fields, Construction D, and Construction $\pi_A$ as special cases.Comment: 26 pages, 11 figures. arXiv admin note: text overlap with arXiv:1401.222

Multistage Compute-and-Forward with Multilevel Lattice Codes Based on Product Constructions

A novel construction of lattices is proposed. This construction can be thought of as Construction A with codes that can be represented as the Cartesian product of $L$ linear codes over $\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$, respectively; hence, is referred to as the product construction. The existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is shown. This family of lattices is then used to generate a sequence of nested lattice codes which allows one to achieve the same computation rate of Nazer and Gastpar for compute-and-forward under multistage decoding, which is referred to as lattice-based multistage compute-and-forward. Motivated by the proposed lattice codes, two families of signal constellations are then proposed for the separation-based compute-and-forward framework proposed by Tunali \textit{et al.} together with a multilevel coding/multistage decoding scheme tailored specifically for these constellations. This scheme is termed separation-based multistage compute-and-forward and is shown having a complexity of the channel coding dominated by the greatest common divisor of the constellation size (may not be a prime number) instead of the constellation size itself.Comment: 45 pages, 22 figure

Construction of Barnes-Wall Lattices from Linear Codes over Rings

Dense lattice packings can be obtained via the wellknown Construction A from binary linear codes. In this paper, we use an extension of Construction A called Construction A ′ to obtain Barnes-Wall lattices from linear codes over polynomials rings. To obtain the Barnes-Wall lattice BW2m in C2m for any m ≥ 1, we first identify a linear code C2m over the quotient ring Um = F2[u]�u m and then propose a mapping ψ: Um → Z[i] such that the code L2m = ψ(C2m) is a lattice constellation. Further, we show that L2m has the cubic shaping property when m is even. Finally, we show that BW2m can be obtained through Construction A ′ as BW2m =(1+i)mZ[i] 2