Analysis of Low-Density Parity-Check Codes over Finite Integer Rings for the Lee Channel

TL;DR

This paper investigates the performance of nonbinary LDPC codes over finite integer rings for channels based on the Lee metric, deriving bounds and analyzing decoding algorithms through simulations.

Contribution

It introduces a detailed analysis of LDPC codes over finite rings for Lee metric channels, including bounds and decoding performance evaluation.

Findings

Marginal conditional distributions coincide for large block lengths.

Derived random coding union bounds for block error probability.

Density evolution and simulations show decoding performance relative to bounds.

Abstract

We study the performance of nonbinary low-density parity-check (LDPC) codes over finite integer rings over two channels that arise from the Lee metric. The first channel is a discrete memory-less channel (DMC) matched to the Lee metric. The second channel adds to each codeword an error vector of constant Lee weight, where the error vector is picked uniformly at random from the set of vectors of constant Lee weight. It is shown that the marginal conditional distributions of the two channels coincide, in the limit of large block length. Random coding union bounds on the block error probability are derived for both channels. Moreover, the performance of selected LDPC code ensembles is analyzed by means of density evolution and finite-length simulations, with belief propagation decoding and with a low-complexity symbol message passing algorithm and it is compared to the derived bounds.