[Paper Review] Recursive decoding of Reed-Muller codes

Ilya Dumer|arXiv (Cornell University)|Mar 14, 2017

Coding theory and cryptography4 references4 citations

TL;DR

This paper presents a recursive soft- and hard-decision decoding algorithm for Reed-Muller (RM) codes using the Plotkin $(u,u+v)$ construction to decompose codes into smaller constituent codes. By recursively decoding at end nodes $inom{j}{1}$ and $inom{j}{j-1}$ while recalculating likelihoods via probabilistic methods, the algorithm achieves near-optimal error correction with $O(n\log n)$ complexity, significantly outperforming prior methods by correcting up to $d\ln d/2$ errors for medium and high-rate codes.

ABSTRACT

New soft- and hard decision decoding algorithms are presented for general Reed-Muller codes $\left\{\genfrac{}{}{0pt}{}{m}{r} ight\} $ of length $2^{m}$ and distance $2^{m-r}$. We use Plotkin $(u,u+v)$ construction and decompose code $\left\{\genfrac{}{}{0pt}{}{m}{r} ight\} $ onto subblocks $u\in\left\{\genfrac{}{}{0pt}{}{m-1}{r} ight\} $ and $v\in\left\{\genfrac{}{}{0pt}{}{m-1}{r-1} ight\} .$ In decoding, we first try to find a subblock $v$ from the better protected code and then proceed with the block $u$. The likelihoods of the received symbols are recalculated in a way similar to belief propagation. Thus, decoding is relegated to the two constituent codes. We repeat this recursion and execute decoding only at the end nodes $\left\{\genfrac{}{}{0pt}{}{j}{1} ight\} $ and $\left\{\genfrac{}{}{0pt}{}{j}{j-1} ight\} $. The overall complexity has low order of $n\log n.$ It is shown that this decoding substantially outperforms other algorithms of polynomial complexity known for RM codes. In particular, for medium and high code rates, the algorithm corrects most error patterns of weight $d\ln d/2.$

Motivation & Objective

• To develop a low-complexity decoding algorithm for Reed-Muller codes that improves upon existing polynomial-time methods.
• To extend recursive decoding beyond bounded-distance decoding by leveraging soft-decision metrics and probabilistic likelihood updates.
• To improve error-correcting performance for medium- and high-rate RM codes, especially in short-to-moderate block lengths.
• To optimize the decoding termination rule by stopping at first-order and $(j-1)$-order codes instead of zero-order codes, enhancing decoding capacity.
• To demonstrate that recursive decoding with probabilistic likelihood recalculations achieves performance close to majority decoding while maintaining low complexity.

Proposed method

• Decomposes RM codes $\binom{m}{r}$ recursively using the Plotkin $(u,u+v)$ construction into subcodes $\binom{m-1}{r}$ and $\binom{m-1}{r-1}$.
• Recalculates likelihoods of received symbols at each recursion level using belief-propagation-like updates to propagate soft information.
• Executes actual decoding only at terminal nodes $\binom{j}{1}$ (biorthogonal codes) and $\binom{j}{j-1}$ (single parity-check codes).
• Uses a recursive decoding strategy that processes the Pascal triangle structure of RM codes, starting from the root and moving toward end nodes.
• Applies an expurgation technique by setting the least protected information bits (e.g., from leftmost end node) to zero, improving subcode performance.
• Employs soft-decision metrics such as squared Euclidean distance $\rho^2$ and hard-decision thresholds $t$ to quantify decoding capacity.

Experimental results

Research questions

• RQ1 Can recursive decoding of Reed-Muller codes be extended beyond bounded-distance decoding to correct more error patterns?
• RQ2 How does probabilistic likelihood recalibration during recursion affect decoding performance compared to traditional recursive algorithms?
• RQ3 What is the optimal stopping rule for recursive decoding in terms of decoding capacity and complexity?
• RQ4 Can subcode optimization via information bit expurgation significantly improve block error rate performance?
• RQ5 How does the new algorithm’s error-correcting capacity compare quantitatively to majority decoding and prior recursive methods?

Key findings

• The algorithm achieves $O(n\log n)$ complexity, significantly lower than prior recursive algorithms with $O(n\min(r,m-r))$ complexity.
• For medium and high-rate codes, the algorithm corrects up to $d\ln d/2$ error patterns, doubling the decoding capacity of previous recursive algorithms.
• In simulations of the $\binom{9}{4}$ code (length 512), the new recursive decoding achieved a block error rate of $2\times10^{-4}$ at 4 dB SNR, outperforming prior recursive and majority decoding methods.
• Expurgating the least protected information bits (from the leftmost end node) reduced block error rate to $2\times10^{-4}$ at 4 dB, showing substantial performance gain.
• The use of first-order and $(j-1)$-order codes as terminal nodes instead of zero-order codes enables higher decoding capacity than previous recursive algorithms.
• The algorithm’s soft-decision performance, measured by $\rho^2$, exceeds that of majority decoding and prior recursive methods, especially for fixed-rate codes.