Multilevel Constructions: Coding, Packing and Geometric Uniformity

Abstract

Lattice and special nonlattice multilevel constellations constructed from binary codes, such as Construction C, have relevant applications in mathematics (sphere packing) and communication problems (multi-stage decoding and efficient vector quantization). In this work, we explore some properties of Construction C, in particular its geometric uniformity. We then propose a new multilevel construction, motivated by bit interleaved coded modulation, that we call Construction C . We explore the geometric uniformity and minimum distance properties of Construction C , and discuss its potential superior packing efficiency with respect to Construction C.

Tiling of Constellations

“…, c 1 ). From this definition, it is clearly valid that 2ψ(c) = ψ(c (1) ). Throughout the paper, we will also refer to the Schur level shift, which means that, given c…”

“…Contributions: In general, Γ C produces a nonlattice constellation, generated by a finite set in Z n 2 L . Recently a subset Γ C ⊆ Γ C was introduced [1], by associating the linear codes underlying both constructions. In this work, we discuss conditions for the existence of a tiling of Γ C by classes of Γ C .…”

“…If (Y, Z) is a tiling of X and at most the second Schur level shift is vanishing, then the Schur level shift (c * d i )(1) ∈ C, for all c ∈ C and d i ∈ D.…”

On the decoding of Barnes-Wall lattices

“…This series includes dense lattices in lower dimensions such as D 4 , E 8 , Λ 16 [5], and is deeply related to Reed-Muller codes [9] [17]: BW lattices admit a Construction D based on these codes. Multilevel constructions attracted the recent attention of researchers, mainly Construction C * [3], where lattice and non-lattice constellations are made out of binary codes. One of the important challenges is to develop lattices with a reasonable-complexity decoding where a fraction of the fundamental coding gain is sacrificed in order to achieve a lower kissing number.…”

Construction $C^\star$ from Self-Dual Codes

“…One way of producing lattice constellations is to use linear codes in the so called Constructions A, B, and D [8]. There are also other interesting constructions that generate more general constellations (lattices and non-lattices) with prominent applications in quantization and coded modulation, such as Constructions C [11] and C ⋆ [4]. The advantage of working with such constructions is mainly the translation of characteristics from the linear code over a finite field to an infinite constellation in the n−dimensional real space.…”