Error-Correction Performance of Regular Ring-Linear LDPC Codes over Lee Channels

TL;DR

The paper investigates regular LDPC codes over finite integer rings under the Lee metric, focusing on two Lee-channel models: a constant Lee weight channel and a memoryless Lee channel whose error marginal follows a Boltzmann distribution. It develops finite-length performance bounds via random coding union and sphere-packing arguments, derives the average Lee weight spectrum for LDPC ensembles, and couples these with density evolution to analyze BP and SMP decoders, validated by finite-length simulations. The results quantify how ring-linear LDPC codes perform in the Lee metric, offering design guidance for storage and cryptographic applications and highlighting the trade-offs between decoding complexity and performance. Overall, the work provides a comprehensive framework linking Lee-weight spectral properties, finite-length bounds, and iterative decoding behavior for LDPC codes over $\mathbb{Z}/q\mathbb{Z}$.

Abstract

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. The analysis developed in this paper may serve to design regular LDPC codes over integer residue rings for storage and cryptographic application.

Figures (14)

• Figure 1: Random coding union bounds under MD decoding based on Theorem \ref{['thm:RCU_const_MD']} and Corollary \ref{['cor:RCU_const_MD']} for the parameters $n=500$ and $k = 250$ over $\mathbb{Z}/{7}\mathbb{Z}$.
• Figure 2: Random coding union (Corollary \ref{['cor:RCU_mless']}) and sphere-packing bounds (Theorem \ref{['thrm:SPB']}) for the parameters $n=1024$ and $k=512$ over $\mathbb{Z}/{7}\mathbb{Z}$.
• Figure 3: Graphical representation of a random $(d_{\mathsf{v}}, d_{\mathsf{c}})$ LDPC code of length $n$.
• Figure 4: Spectral growth rate of the average weight enumerator of a regular $(3, 6)$ LDPC code ensembles over $\mathbb{Z}/{2}\mathbb{Z}$ (solid blue line) versus the spectral growth rate of the average weight enumerator of a random code over $\mathbb{Z}/{2}\mathbb{Z}$ and rate $R=1/2$ (dashed red line). The logarithm is in base $q$.
• Figure 5: Spectral growth rate of the average weight enumerator of a regular $(3, 6)$ LDPC code ensembles over $\mathbb{Z}/{3}\mathbb{Z}$ (solid blue line) versus the spectral growth rate of the average weight enumerator of a random code over $\mathbb{Z}/{3}\mathbb{Z}$ and rate $R=1/2$ (dashed red line). The logarithm is in base $q$.

Theorems & Definitions (48)

• Definition 2.1
• Definition 2.2
• Lemma 2.3: wyner1968upper
• proof
• Lemma 2.4
• Theorem 2.5
• Lemma 3.1: Growth rate of the surface spectrum
• proof
• Lemma 3.2: Growth rate of the volume spectrum
• proof