Most low-density parity-check (LDPC) code constructions are considered over finite fields. In this work, we focus on regular LDPC codes over integer residue rings and analyze their performance with respect to the Lee metric. Their error-correction performance is studied over two channel models, in the Lee metric. The first channel model is a discrete memoryless channel, whereas in the second channel model an error vector is drawn uniformly at random from all vectors of a fixed Lee weight. It is known that the two channel laws coincide in the asymptotic regime, meaning that their marginal distributions match. For both channel models, we derive upper bounds on the block error probability in terms of a random coding union bound as well as sphere packing bounds that make use of the marginal distribution of the considered channels. We estimate the decoding error probability of regular LDPC code ensembles over the channels using the marginal distribution and determining the expected Lee weight distribution of a random LDPC code over a finite integer ring. By means of density evolution and finite-length simulations, we estimate the error-correction performance of selected LDPC code ensembles under belief propagation decoding and a low-complexity symbol message passing decoding algorithm and compare the performances.

翻译：大多数低密度奇偶校验（LDPC）码的构造都基于有限域。本文聚焦于整数剩余环上的正则LDPC码，并分析其在Lee度量下的性能。我们研究了两种Lee度量信道模型下的纠错性能。第一种信道模型是离散无记忆信道，而第二种信道模型中，误差向量从所有具有固定Lee权重的向量中均匀随机抽取。已知这两种信道法则在渐近情况下一致，即它们的边缘分布相匹配。针对这两种信道模型，我们利用随机编码联合界以及基于所考虑信道边缘分布的球堆积界，推导了分块错误概率的上界。我们通过边缘分布估计正则LDPC码集合在该信道上的译码错误概率，并确定有限整数环上随机LDPC码的期望Lee权重分布。通过密度演进和有限长度仿真，我们评估了选定LDPC码集合在置信传播译码和低复杂度符号消息传递译码算法下的纠错性能，并比较了二者的性能。