In this paper, we revisit the Recursive Projection-Aggregation (RPA) decoder, of Ye and Abbe (2020), for Reed-Muller (RM) codes. Our main contribution is an explicit upper bound on the probability of incorrect decoding, using the RPA decoder, over a binary symmetric channel (BSC). Importantly, we focus on the events where a \emph{single} iteration of the RPA decoder, in each recursive call, is sufficient for convergence. Key components of our analysis are explicit estimates of the probability of incorrect decoding of first-order RM codes using a maximum likelihood (ML) decoder, and estimates of the error probabilities during the aggregation phase of the RPA decoder. Our results allow us to show that for RM codes with blocklength $N = 2^m$ , the RPA decoder can achieve vanishing error probabilities, in the large blocklength limit, for RM orders that grow roughly logarithmically in $m$ .In this paper, we revisit the Recursive Projection-Aggregation (RPA) decoder, of Ye and Abbe (2020), for Reed-Muller (RM) codes. Our main contribution is an explicit upper bound on the probability of incorrect decoding, using the RPA decoder, over a binary symmetric channel (BSC). Importantly, we focus on the events where a \emph{single} iteration of the RPA decoder, in each recursive call, is sufficient for convergence. Key components of our analysis are explicit estimates of the probability of incorrect decoding of first-order RM codes using a maximum likelihood (ML) decoder, and estimates of the error probabilities during the aggregation phase of the RPA decoder. Our results allow us to show that for RM codes with blocklength $N = 2^m$, the RPA decoder can achieve vanishing error probabilities, in the large blocklength limit, for RM orders that grow roughly logarithmically in $m$.

arXiv:2412.08129Dec 11, 20240 citations

On the Error Probability of RPA Decoding of Reed-Muller Codes over BMS Channels

Dorsa Fathollahi, V. Arvind Rameshwar, V. Lalitha

We analyze the performance of the Recursive Projection-Aggregation (RPA) decoder of Ye and Abbe (2020), for Reed-Muller (RM) codes, over general binary memoryless symmetric (BMS) channels. Our work is a significant generalization of a recent result of Rameshwar and Lalitha (2025) that showed that the RPA decoder provably achieves vanishing error probabilities for "low-rate" RM codes, over the binary symmetric channel (BSC). While a straightforward generalization of the proof strategy in that paper will require additional, restrictive assumptions on the BMS channel, our technique, which employs an equivalence between the RPA projection operation and a part of the "channel combining" phase in polar codes, requires no such assumptions. Interestingly, such an equivalence allows for the use of a generic union bound on the error probability of the first-order RM code (the "base case" of the RPA decoder), under maximum-likelihood decoding, which holds for any BMS channel. We then exploit these observations in the proof strategy outlined in the work of Rameshwar and Lalitha (2025), and argue that, much like in the case of the BSC, one can obtain vanishing error probabilities, in the large $n$ limit (where $n$ is the blocklength), for RM orders that scale roughly as $\log \log n$ , for all BMS channels.We analyze the performance of the Recursive Projection-Aggregation (RPA) decoder of Ye and Abbe (2020), for Reed-Muller (RM) codes, over general binary memoryless symmetric (BMS) channels. Our work is a significant generalization of a recent result of Rameshwar and Lalitha (2025) that showed that the RPA decoder provably achieves vanishing error probabilities for "low-rate" RM codes, over the binary symmetric channel (BSC). While a straightforward generalization of the proof strategy in that paper will require additional, restrictive assumptions on the BMS channel, our technique, which employs an equivalence between the RPA projection operation and a part of the "channel combining" phase in polar codes, requires no such assumptions. Interestingly, such an equivalence allows for the use of a generic union bound on the error probability of the first-order RM code (the "base case" of the RPA decoder), under maximum-likelihood decoding, which holds for any BMS channel. We then exploit these observations in the proof strategy outlined in the work of Rameshwar and Lalitha (2025), and argue that, much like in the case of the BSC, one can obtain vanishing error probabilities, in the large $n$ limit (where $n$ is the blocklength), for RM orders that scale roughly as $\log \log n$, for all BMS channels.

arXiv:2601.09581IEEE International Symposium on Information Theory (ISIT)Jun 1, 20261 citation

Reed-Muller Subcodes: Machine Learning-Aided Design of Efficient Soft Recursive Decoding

Mohammad Vahid Jamali, Xiyang Liu, Ashok Vardhan Makkuva +3 more

Reed-Muller (RM) codes are conjectured to achieve the capacity of any binary-input memoryless symmetric (BMS) channel, and are observed to have a comparable performance to that of random codes in terms of scaling laws. On the negative side, RM codes lack efficient decoders with performance close to that of a maximum likelihood decoder for general parameters. Also, they only admit certain discrete sets of rates. In this paper, we focus on subcodes of RM codes with flexible rates that can take any code dimension from 1 to $n$ . where $n$ is the blocklength. We first extend the recursive projection-aggregation (RPA) algorithm proposed recently by Ye and Abbe for decoding RM codes. To lower the complexity of our decoding algorithm, referred to as subRPA, we investigate different ways for pruning the projections. We then derive the soft-decision based version of our algorithm, called soft-subRPA, that is shown to improve upon the performance of subRPA. Furthermore, it enables training a machine learning (ML) model to search for good sets of projections that minimize the decoding error rate. Training our ML model enables achieving very close to the performance of full-projection decoding with a significantly reduced number of projections. For instance, our simulation results on a (64,14) RM subcode show almost identical performance for full-projection decoding and pruned-projection decoding with 15 projections picked via training our ML model. This is equivalent to lowering the complexity by a factor of more than 4 without sacrificing the decoding performance.Reed-Muller (RM) codes are conjectured to achieve the capacity of any binary-input memoryless symmetric (BMS) channel, and are observed to have a comparable performance to that of random codes in terms of scaling laws. On the negative side, RM codes lack efficient decoders with performance close to that of a maximum likelihood decoder for general parameters. Also, they only admit certain discrete sets of rates. In this paper, we focus on subcodes of RM codes with flexible rates that can take any code dimension from 1 to$n$. where$n$is the blocklength. We first extend the recursive projection-aggregation (RPA) algorithm proposed recently by Ye and Abbe for decoding RM codes. To lower the complexity of our decoding algorithm, referred to as subRPA, we investigate different ways for pruning the projections. We then derive the soft-decision based version of our algorithm, called soft-subRPA, that is shown to improve upon the performance of subRPA. Furthermore, it enables training a machine learning (ML) model to search for good sets of projections that minimize the decoding error rate. Training our ML model enables achieving very close to the performance of full-projection decoding with a significantly reduced number of projections. For instance, our simulation results on a (64,14) RM subcode show almost identical performance for full-projection decoding and pruned-projection decoding with 15 projections picked via training our ML model. This is equivalent to lowering the complexity by a factor of more than 4 without sacrificing the decoding performance.

arXiv:2102.01671IEEE International Symposium on Information Theory (ISIT)Jul 12, 202116 citations

Decoding Reed–Muller Codes Using Redundant Code Constraints

Mengke Lian, Christian Häger, Henry D. Pfister

The recursive projection-aggregation (RPA) decoding algorithm for Reed-Muller (RM) codes was recently introduced by Ye and Abbe. We show that the RPA algorithm is closely related to (weighted) belief-propagation (BP) decoding by interpreting it as a message-passing algorithm on a factor graph with redundant code constraints. We use this observation to introduce a novel decoder tailored to high-rate RM codes. The new algorithm relies on puncturing rather than projections and is called recursive puncturing-aggregation (RXA). We also investigate collapsed (i.e., non-recursive) versions of RPA and RXA and show some examples where they achieve similar performance with lower decoding complexity.The recursive projection-aggregation (RPA) decoding algorithm for Reed-Muller (RM) codes was recently introduced by Ye and Abbe. We show that the RPA algorithm is closely related to (weighted) belief-propagation (BP) decoding by interpreting it as a message-passing algorithm on a factor graph with redundant code constraints. We use this observation to introduce a novel decoder tailored to high-rate RM codes. The new algorithm relies on puncturing rather than projections and is called recursive puncturing-aggregation (RXA). We also investigate collapsed (i.e., non-recursive) versions of RPA and RXA and show some examples where they achieve similar performance with lower decoding complexity.

IEEE International Symposium on Information Theory (ISIT)Jun 1, 202045 citations

Multi-Factor Pruning for Recursive Projection-Aggregation Decoding of RM Codes

Marzieh Hashemipour-Nazari, Kees G. W. Goossens, Alexios Balatsoukas‐Stimming

The recently introduced recursive projection aggregation (RPA) decoding method for Reed-Muller (RM) codes can achieve near-maximum likelihood (ML) decoding performance. However, its high computational complexity makes its implementation challenging for time- and resource-critical applications. In this work, we present a complexity reduction technique called multi-factor pruning that reduces the computational complexity of RPA significantly. Our simulation results show that the proposed pruning approach with appropriately selected factors can reduce the complexity of RPA by up to 92% for RM(8, 3) while keeping a comparable error-correcting performance.The recently introduced recursive projection aggregation (RPA) decoding method for Reed-Muller (RM) codes can achieve near-maximum likelihood (ML) decoding performance. However, its high computational complexity makes its implementation challenging for time- and resource-critical applications. In this work, we present a complexity reduction technique called multi-factor pruning that reduces the computational complexity of RPA significantly. Our simulation results show that the proposed pruning approach with appropriately selected factors can reduce the complexity of RPA by up to 92% for RM(8, 3) while keeping a comparable error-correcting performance.

arXiv:2208.13659SiPS ... design and implementation - IEEE Workshop on Signal Processing SystemsNov 2, 20228 citations

Machine Learning-Aided Efficient Decoding of Reed–Muller Subcodes

Mohammad Vahid Jamali, Xiyang Liu, Ashok Vardhan Makkuva +3 more

Reed-Muller (RM) codes achieve the capacity of general binary-input memoryless symmetric channels and are conjectured to have a comparable performance to that of random codes in terms of scaling laws. However, such results are established assuming maximum-likelihood decoders for general code parameters. Also, RM codes only admit limited sets of rates. Efficient decoders such as successive cancellation list (SCL) decoder and recently-introduced recursive projection-aggregation (RPA) decoders are available for RM codes at finite lengths. In this paper, we focus on subcodes of RM codes with flexible rates. We first extend the RPA decoding algorithm to RM subcodes. To lower the complexity of our decoding algorithm, referred to as subRPA, we investigate different approaches to prune the projections. Next, we derive the soft-decision based version of our algorithm, called soft-subRPA, that not only improves upon the performance of subRPA but also enables a differentiable decoding algorithm. Building upon the soft-subRPA algorithm, we then provide a framework for training a machine learning (ML) model to search forgoodsets of projections that minimize the decoding error rate. Training our ML model enables achieving very close to the performance of full-projection decoding with a significantly smaller number of projections. We also show that the choice of the projections in decoding RM subcodes matters significantly, and our ML-aided projection pruning scheme is able to find agoodselection, i.e., with negligible performance degradation compared to the full-projection case, given a reasonable number of projections.Reed-Muller (RM) codes achieve the capacity of general binary-input memoryless symmetric channels and are conjectured to have a comparable performance to that of random codes in terms of scaling laws. However, such results are established assuming maximum-likelihood decoders for general code parameters. Also, RM codes only admit limited sets of rates. Efficient decoders such as successive cancellation list (SCL) decoder and recently-introduced recursive projection-aggregation (RPA) decoders are available for RM codes at finite lengths. In this paper, we focus on subcodes of RM codes with flexible rates. We first extend the RPA decoding algorithm to RM subcodes. To lower the complexity of our decoding algorithm, referred to as subRPA, we investigate different approaches to prune the projections. Next, we derive the soft-decision based version of our algorithm, called soft-subRPA, that not only improves upon the performance of subRPA but also enables a differentiable decoding algorithm. Building upon the soft-subRPA algorithm, we then provide a framework for training a machine learning (ML) model to search forgoodsets of projections that minimize the decoding error rate. Training our ML model enables achieving very close to the performance of full-projection decoding with a significantly smaller number of projections. We also show that the choice of the projections in decoding RM subcodes matters significantly, and our ML-aided projection pruning scheme is able to find agoodselection, i.e., with negligible performance degradation compared to the full-projection case, given a reasonable number of projections.

arXiv:2301.06251IEEE Journal on Selected Areas in Information TheoryJan 1, 20232 citations

N=2^{m}

, the RPA decoder can achieve vanishing error probabilities, in the large blocklength limit, for RM orders that grow roughly logarithmically in

m

N = 2^m

, the RPA decoder can achieve vanishing error probabilities, in the large blocklength limit, for RM orders that grow roughly logarithmically in

m

n

limit (where

n

is the blocklength), for RM orders that scale roughly as

\log \log n

, for all BMS channels.

n

. where

n

is the blocklength. We first extend the recursive projection-aggregation (RPA) algorithm proposed recently by Ye and Abbe for decoding RM codes. To lower the complexity of our decoding algorithm, referred to as subRPA, we investigate different ways for pruning the projections. We then derive the soft-decision based version of our algorithm, called soft-subRPA, that is shown to improve upon the performance of subRPA. Furthermore, it enables training a machine learning (ML) model to search for good sets of projections that minimize the decoding error rate. Training our ML model enables achieving very close to the performance of full-projection decoding with a significantly reduced number of projections. For instance, our simulation results on a (64,14) RM subcode show almost identical performance for full-projection decoding and pruned-projection decoding with 15 projections picked via training our ML model. This is equivalent to lowering the complexity by a factor of more than 4 without sacrificing the decoding performance.