Efficient Soft-Decision Decoding of Linear Block Codes with Reduced Gaussian Elimination Complexity

Core Concepts

A modified OSD algorithm is presented that performs a limited Gaussian Elimination (GE) with reduced complexity of O(N^3 min{R, 1-R}^3) for an (N, K) linear block code of rate R = K/N.

Abstract

The paper presents a modified version of the Ordered Statistic Decoding (OSD) algorithm that reduces the complexity of the Gaussian Elimination (GE) step. The key ideas are: Separating the information positions from the parity positions in both the Most Reliable Basis (MRB) and the Least Reliable Basis (LRB). This allows avoiding GE on certain parts of the generator matrix. Applying a two-stage decoding approach, where the first stage processes the most reliable information positions and the second stage processes the less reliable information positions and parity positions. Further generalizing this approach to a multi-stage decoding, where the GE complexity can be reduced even more by bounding the number of reprocessed rows. The proposed modifications achieve significant complexity savings compared to the original OSD algorithm, while maintaining near-optimal Maximum Likelihood Decoding (MLD) performance. Simulation results for BCH codes demonstrate the effectiveness of the proposed techniques.

Stats

The complexity of the full GE is O(N^3 min{R, 1-R}^2). The complexity of the two-stage reduced GE is O(N^3 min{R, 1-R}^3). The complexity of the three-stage reduced GE with optimized parameter α is O((N-|BK,MR|) * (|BK,LR| - α) * |BK,LR| + (N-K+α) * α^2).

Quotes

"The main idea is to separate the information positions from the parity positions of the original representation of the code in both the MRB and its dual least reliable basis (LRB). As a result the original information positions also in the MRB do not need to be considered by the GE in G-space decoding. Similarly the original parity positions also in the LRB do not need to be considered by the GE in H-space decoding." "Consequently OSD based on G′2 in (3) is achieved with a two stage decoding."

Key Insights Distilled From

Modified OSD Algorithm with Reduced Gaussian Elimination

by Marc Fossori... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.02293.pdf

Deeper Inquiries

How can the proposed techniques be extended to other soft-decision decoding algorithms beyond OSD

The proposed techniques in the modified OSD algorithm can be extended to other soft-decision decoding algorithms by adapting the concept of reduced Gaussian elimination (GE) complexity and multi-stage decoding. For instance, in algorithms like belief propagation (BP) or sum-product decoding, the reduced GE approach can help in simplifying the decoding process by minimizing the number of dependency occurrences and optimizing the systematic form of the code matrix. By incorporating the idea of separating information and parity positions, similar efficiency gains can be achieved in various soft-decision decoding algorithms.

What are the potential hardware implementation advantages of the reduced GE complexity in the modified OSD algorithm

The reduced GE complexity in the modified OSD algorithm offers several hardware implementation advantages. Firstly, by reducing the computational complexity from cubic to cubic-linear complexity, the hardware resources required for GE processing are significantly minimized. This reduction in complexity leads to faster decoding speeds and lower power consumption, making it more suitable for real-time applications and resource-constrained devices. Additionally, the optimized systematic form obtained through reduced GE simplifies the hardware architecture, enabling efficient parallel processing and reducing memory requirements. Overall, the hardware implementation benefits include improved performance, reduced area utilization, and enhanced energy efficiency.

Can the multi-stage decoding approach be further optimized to find the best trade-off between complexity and performance for different code parameters and channel conditions

The multi-stage decoding approach in the modified OSD algorithm can be further optimized to find the best trade-off between complexity and performance for different code parameters and channel conditions. By adjusting the number of stages and the size of the reduced GE matrices based on specific code characteristics such as block length, rate, and error-correction capability, the decoding complexity can be tailored to achieve the desired balance between computational efficiency and decoding accuracy. Through empirical analysis and optimization techniques, the optimal configuration of stages, row limits, and column permutations can be determined to maximize decoding performance while minimizing computational overhead. This adaptive optimization process ensures that the multi-stage decoding approach is finely tuned to deliver the best results for a wide range of code parameters and channel scenarios.