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Construction A Lattice Design Based on the Truncated Union Bound

Jiajie Xue, Brian M. Kurkoski, Emanuele Viterbo

TL;DR

The paper addresses finite‑dimensional Construction A lattice design at $n=128$ by optimizing binary component codes using a truncated union bound derived from a truncated theta series. It selects the component code $\mathcal{C}$ to minimize the required $\text{VNR}$ for a target word error rate $P_e$, and demonstrates results with EBCH and polar codes showing a new best known lattice at $P_e=10^{-5}$ using $(128,106,8)$ EBCH, as well as strong performance from polar‑code lattices (e.g., $(128,99,8)$). The approach provides analytic design guidance with lower decoding complexity than Construction D, validated for $P_e\le 10^{-4}$ and extendable to higher dimensions and other binary codes; the work highlights practical near‑ML performance for medium‑dimensional lattice codes and its potential impact on compact, low‑latency communications.

Abstract

This paper considers $n= 128$ dimensional construction A lattice design, using binary codes with known minimum Hamming distance and codeword multiplicity, the number of minimum weight codeword. A truncated theta series of the lattice is explicitly given to obtain the truncated union bound to estimate the word error rate under maximum likelihood decoding. The best component code is selected by minimizing the required volume-to-noise ratio (VNR) for a target word error rate $P_e$. The estimate becomes accurate for $P_e \leq 10^{-4}$, and design examples are given with the best extended BCH codes and polar codes for $P_e= 10^{-4}$ to $10^{-8}$. A lower error rate is achieved compared to that by the classic balanced distance rule and the equal error probability rule. The $(128, 106, 8)$ EBCH code gives the best-known $n=128$ construction A lattice at $P_e= 10^{-5}$.

Construction A Lattice Design Based on the Truncated Union Bound

TL;DR

The paper addresses finite‑dimensional Construction A lattice design at by optimizing binary component codes using a truncated union bound derived from a truncated theta series. It selects the component code to minimize the required for a target word error rate , and demonstrates results with EBCH and polar codes showing a new best known lattice at using EBCH, as well as strong performance from polar‑code lattices (e.g., ). The approach provides analytic design guidance with lower decoding complexity than Construction D, validated for and extendable to higher dimensions and other binary codes; the work highlights practical near‑ML performance for medium‑dimensional lattice codes and its potential impact on compact, low‑latency communications.

Abstract

This paper considers dimensional construction A lattice design, using binary codes with known minimum Hamming distance and codeword multiplicity, the number of minimum weight codeword. A truncated theta series of the lattice is explicitly given to obtain the truncated union bound to estimate the word error rate under maximum likelihood decoding. The best component code is selected by minimizing the required volume-to-noise ratio (VNR) for a target word error rate . The estimate becomes accurate for , and design examples are given with the best extended BCH codes and polar codes for to . A lower error rate is achieved compared to that by the classic balanced distance rule and the equal error probability rule. The EBCH code gives the best-known construction A lattice at .

Paper Structure

This paper contains 11 sections, 1 theorem, 25 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices
  4. Construction A lattices
  5. Encoding and decoding construction A lattices
  6. Lattice design with the truncated union bound
  7. Design examples using binary codes
  8. Design using extended BCH codes
  9. Design using polar codes
  10. Comparison with different design rules
  11. Conclusion

Key Result

Proposition 1

rowshan2023formation Consider an $(n, k, d_c)$ polar code satisfy the partial order property in rowshan2023formation. For all $i \in \mathcal{I}'$, let where $\rm{wt}(\mathbf{g}_i)$ is the weight of the $i$-th row of the polar transformation matrix. The codeword multiplicity is found by:

Figures (5)

  • Figure 1: Encoder and decoder of construction A lattices.
  • Figure 2: The truncated union bound estimate and numerical evaluation of $P_e$ for EBCH code lattice with $d_c= 4, 6, 8, 10$. Order-2 OSD algorithm is used to decode EBCH codes.
  • Figure 3: Truncated union bound of construction A lattices using polar codes with different $\tau_c$ for $d_c= 4, 8$.
  • Figure 4: The truncated union bound estimate and numerical evaluation of $P_e$ for polar code lattice with different code rates for $d_c= 4, 8, 16$. Order-2 OSD algorithm is used to decode polar codes.
  • Figure 5: WER performance for construction A lattices with different design rules. Order-2 OSD is used to decode component codes of construction A lattices. Construction D lattices are from matsumine2018construction and ludwiniananda2021design.

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Proposition 1