Construction A Lattice Design Based on the Truncated Union Bound

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 $P_e= 10^{-5}$.