Low-Dimensional Shaping for High-Dimensional Lattice Codes

Abstract

We propose two low-complexity lattice code constructions that have competitive coding and shaping gains. The first construction, named systematic Voronoi shaping, maps short blocks of integers to the dithered Voronoi integers, which are dithered integers that are uniformly distributed over the Voronoi region of a low-dimensional shaping lattice. Then, these dithered Voronoi integers are encoded using a high-dimensional lattice retaining the same shaping and coding gains of low and highdimensional lattices. A drawback to this construction is that there is no isomorphism between the underlying message and the lattice code, preventing its use in applications such as computeand-forward. Therefore we propose a second construction, called mixed nested lattice codes, in which a high-dimensional coding lattice is nested inside a concatenation of low-dimensional shaping lattices. This construction not only retains the same shaping/coding gains as first construction but also provides the desired algebraic structure. We numerically study these methods, for point-to-point channels as well as compute-and-forward using low-density lattice codes (LDLCs) as coding lattices and E8 and Barnes-Wall as shaping lattices. Numerical results indicate a shaping gain of up to 0.86 dB, compared to the state-of-the-art of 0.4 dB; furthermore, the proposed method has lower complexity than state-of-the-art approaches.

Construction D' Lattices for Power-Constrained Communications

“…A pair of distinct lattices Λ c and Λ s to form C is desirable, where Λ c is good for coding and Λ s is good for shaping. This was used in past work: shaping LDLC lattices using the E 8 lattice and the BW 16 lattice [20], convolutional code lattices [11], and shaping LDA lattices using the Leech lattice [21]. These results show that Λ c and Λ s can be designed to provide both good coding properties and efficiently achievable shaping gains.…”

“…Using self-similar nested lattice codes, a shaping gain of 0.4 dB was shown for low-density lattice codes (LDLCs) [18], and a shaping gain of 0.776 dB was claimed at n " 60 for Construction A lattices based on QC-LDPC codes [19]. A shaping gain of 0.65 dB and 0.86 dB was observed using the E 8 lattice and the BW 16 lattice for shaping LDLC lattices, respectively [20]. Leech lattice has 1.03 dB shaping gain, and was used to shape LDA lattices [21].…”

“…An improvement for around 0.03-0.08 dB shaping gain can be obtained using generator polynomials p77, 76, 73q at n " 18 and p331, 257q at n " 24 for zerotailed convolutional codes, and using generator polynomials p31, 27q, p73, 25q, p250, 67q and p37, 33, 25q at n " 144 for tail-biting convolutional codes instead. The ultimate shaping gain obtained at n " 2 20 and n " 2 20 `2 for rate 1{2 and 1{3 convolutional codes respectively is also provided. It is observed that at n ď 1152 the shaping gain of convolutional code lattices using tail-biting convolutional codes can achieve the ultimate shaping gain.…”

Low-complexity Voronoi shaping for the Gaussian channel

“…In order to make bit mapping possible, both M and m must be powers of 2. With m an arbitrary power of 2, Λ/mΛ s gives a resolution of 2 bits/symbol/dimension-pair according to (20). However, Λ/mΛ s R offers the intermediate spectral efficiencies as in (23), which decreases the overall resolution to 1 bit/symbol/dimension-pair.…”

“…Ferdinand et al proposed a two-step VC construction method called "systematic Voronoi shaping" in [20] based on the concept of "systematic shaping" proposed by Sommer et al in [21], combining a high-dimensional coding lattice defined by a lower-triangular parity check matrix and a lowerdimensional shaping lattice, to achieve high coding gains and high shaping gains. The SER performance was evaluated when low-density lattice codes [22] are used as the coding lattice and some common multidimensional lattices with low-complexity quantization algorithms are used for the shaping lattice.…”