[Paper Review] Turbo Lattices: Construction and Error Decoding Performance

Amin Sakzad, Mohammad‐Reza Sadeghi+1 more|arXiv (Cornell University)|Aug 9, 2011

Advanced Wireless Communication TechniquesEngineering31 references17 citations

TL;DR

This paper introduces turbo lattices, a new class of lattices constructed via Construction D using nested turbo codes with tail-biting and zero-tail convolutional codes. The proposed multi-stage iterative decoding algorithm achieves near-capacity performance, with only ~1.25 dB from capacity at block length n=1035 and ~0.5 dB at n=10131 for SER=10⁻⁵.

ABSTRACT

In this paper a new class of lattices called turbo lattices is introduced and established. We use the lattice Construction D to produce turbo lattices. This method needs a set of nested linear codes as its underlying structure. We benefit from turbo codes as our basis codes. Therefore, a set of nested turbo codes based on nested interleavers (block interleavers) and nested convolutional codes is built. To this end, we employ both tail-biting and zero-tail convolutional codes. Using these codes, along with construction D, turbo lattices are created. Several properties of Construction D lattices and fundamental characteristics of turbo lattices including the minimum distance, coding gain and kissing number are investigated. Furthermore, a multi-stage turbo lattice decoding algorithm based on iterative turbo decoding algorithm is given. We show, by simulation, that turbo lattices attain good error performance within $\sim1.25 dB$ from capacity at block length of $n=1035$. Also an excellent performance of only $\sim.5 dB$ away from capacity at SER of $10^{-5}$ is achieved for size $n=10131$.

Motivation & Objective

• To develop a new class of lattices, termed turbo lattices, by leveraging the structure of turbo codes.
• To investigate fundamental lattice parameters such as minimum distance, coding gain, and kissing number in the context of turbo lattices.
• To design a multi-stage decoding algorithm based on iterative turbo decoding for efficient lattice decoding.
• To evaluate the error performance of turbo lattices and compare them with existing LDPC and LDLC lattices.

Proposed method

• Constructs turbo lattices using Construction D with a set of nested linear codes as the underlying structure.
• Employs nested interleavers and both tail-biting and zero-tail convolutional codes to build nested turbo codes.
• Applies generalized min-sum and sum-product algorithms to decode the lattice, adapted to the turbo structure.
• Develops a multi-stage decoding algorithm that iteratively decodes lattice components using the iterative nature of turbo decoding.
• Derives upper bounds on the kissing number of Construction D lattices by analyzing minimum-weight codewords in constituent codes.
• Generalizes the minimum distance formula for Construction D lattices by removing restrictive assumptions on constituent code distances.

Experimental results

Research questions

• RQ1 How can turbo codes be adapted to construct lattices with high coding gain and low decoding complexity?
• RQ2 What are the fundamental properties—minimum distance, coding gain, kissing number—of lattices constructed via Construction D using turbo codes?
• RQ3 Can a multi-stage iterative decoding algorithm be designed that leverages the turbo structure for efficient lattice decoding?
• RQ4 How close to capacity do turbo lattices perform in terms of SNR at practical block lengths?
• RQ5 What is the impact of using tail-biting versus zero-tail convolutional codes on lattice performance and rate efficiency?

Key findings

• Turbo lattices achieve approximately 1.25 dB from capacity at block length n=1035.
• For block length n=10131, turbo lattices achieve only ~0.5 dB from capacity at a SER of 10⁻⁵.
• The coding gain of turbo lattices is derived as γ(Λ_TC) = 4^(∑R_ℓ - 1) × min{4, d_min^(ℓ)/(4^(ℓ-1))}, showing dependence on constituent code rates and distances.
• An upper bound on the kissing number is derived as |2n + Σ 2^(d_min^(ℓ)) × A_d_min^(ℓ)| for all ℓ where d_min^(ℓ) ≤ 4^ℓ.
• The multi-stage decoding algorithm ensures correct decoding when the received signal power is below a threshold related to the minimum distance of the constituent codes.
• The minimum distance of the turbo lattice is bounded by d_min²(Λ₀) = 4^(a-1) × min{4, d_min^(ℓ)/(4^(ℓ-1))}, demonstrating the role of component code distances.