# KO codes: Inventing Nonlinear Encoding and Decoding for Reliable Wireless Communication via Deep-learning

arXiv: https://arxiv.org/abs/2108.12920 · HTML: https://mlsanity.com/paper/2108.12920/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 65 references that are in the map.
- First posted on arXiv 29 Aug 2021.

## Builds on (most cited first, 19 of 19)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [An Introduction to Deep Learning for the Physical Layer](https://mlsanity.com/paper/1702.00832/) (2017)
- [Deep Learning-Based Communication Over the Air](https://mlsanity.com/paper/1707.03384/) (2017)
- [Deep Learning Methods for Improved Decoding of Linear Codes](https://mlsanity.com/paper/1706.07043/) (2017)
- [Learning to Decode Linear Codes Using Deep Learning](https://mlsanity.com/paper/1607.04793/) (2016)
- [Turbo Autoencoder: Deep learning based channel codes for point-to-point communication channels](https://mlsanity.com/paper/1911.03038/) (2019)
- [Communication Algorithms via Deep Learning](https://mlsanity.com/paper/1805.09317/) (2018)
- [Hyper-Graph-Network Decoders for Block Codes](https://mlsanity.com/paper/1909.09036/) (2019)
- [Deep Learning for Decoding of Linear Codes - A Syndrome-Based Approach](https://mlsanity.com/paper/1802.04741/) (2018)
- [Learning to Communicate: Channel Auto-encoders, Domain Specific Regularizers, and Attention](https://mlsanity.com/paper/1608.06409/) (2016)
- [Cyclically Equivariant Neural Decoders for Cyclic Codes](https://mlsanity.com/paper/2105.05540/) (2021)
- [Reinforcement Learning for Channel Coding: Learned Bit-Flipping Decoding](https://mlsanity.com/paper/1906.04448/) (2019)
- [Deep Learning-based Polar Code Design](https://mlsanity.com/paper/1909.12035/) (2019)
- [Pruning Neural Belief Propagation Decoders](https://mlsanity.com/paper/2001.07464/) (2020)
- [LEARN Codes: Inventing Low-latency Codes via Recurrent Neural Networks](https://mlsanity.com/paper/1811.12707/) (2018)
- [Learned Belief-Propagation Decoding with Simple Scaling and SNR Adaptation](https://mlsanity.com/paper/1901.08621/) (2019)
- [DeepTurbo: Deep Turbo Decoder](https://mlsanity.com/paper/1903.02295/) (2019)
- [Low-complexity Recurrent Neural Network-based Polar Decoder with Weight Quantization Mechanism](https://mlsanity.com/paper/1810.12154/) (2018)
- [Learning to Flip Successive Cancellation Decoding of Polar Codes with LSTM Networks](https://mlsanity.com/paper/1902.08394/) (2019)

## Built on by (most cited first, 17 of 17)

- [ProductAE: Towards Training Larger Channel Codes based on Neural Product Codes](https://mlsanity.com/paper/2110.04466/) (2021)
- [Semantic Communications with Discrete-time Analog Transmission: A PAPR Perspective](https://mlsanity.com/paper/2208.08342/) (2022)
- [Concatenated Classic and Neural (CCN) Codes: ConcatenatedAE](https://mlsanity.com/paper/2209.01701/) (2022)
- [CRISP: Curriculum based Sequential Neural Decoders for Polar Code Family](https://mlsanity.com/paper/2210.00313/) (2022)
- [Robust Non-Linear Feedback Coding via Power-Constrained Deep Learning](https://mlsanity.com/paper/2304.13178/) (2023)
- [TinyTurbo: Efficient Turbo Decoders on Edge](https://mlsanity.com/paper/2209.15614/) (2022)
- [Data-Driven Neural Polar Codes for Unknown Channels With and Without Memory](https://mlsanity.com/paper/2309.03148/) (2023)
- [Nested Construction of Polar Codes via Transformers](https://mlsanity.com/paper/2401.17188/) (2024)
- [Generalization Bounds for Transformer Channel Decoders](https://mlsanity.com/paper/2601.06969/) (2026)
- [Optimizing Serially Concatenated Neural Codes with Classical Decoders](https://mlsanity.com/paper/2212.10355/) (2022)
- [Machine Learning-Aided Efficient Decoding of Reed-Muller Subcodes](https://mlsanity.com/paper/2301.06251/) (2023)
- [ProductAE: Toward Deep Learning Driven Error-Correction Codes of Large Dimensions](https://mlsanity.com/paper/2303.16424/) (2023)
- [Friendly Attacks to Improve Channel Coding Reliability](https://mlsanity.com/paper/2401.14184/) (2024)
- [DeepPolar: Inventing Nonlinear Large-Kernel Polar Codes via Deep Learning](https://mlsanity.com/paper/2402.08864/) (2024)
- [Interpreting Training Aspects of Deep-Learned Error-Correcting Codes](https://mlsanity.com/paper/2305.04347/) (2023)
- [Disturbance-based Discretization, Differentiable IDS Channel, and an IDS-Correcting Code for DNA-based Storage](https://mlsanity.com/paper/2407.18929/) (2024)
- [Learning Linear Block Codes with Gradient Quantization](https://mlsanity.com/paper/2503.16169/) (2025)

## Abstract (arXiv)

Landmark codes underpin reliable physical layer communication, e.g., Reed-Muller, BCH, Convolution, Turbo, LDPC and Polar codes: each is a linear code and represents a mathematical breakthrough. The impact on humanity is huge: each of these codes has been used in global wireless communication standards (satellite, WiFi, cellular). Reliability of communication over the classical additive white Gaussian noise (AWGN) channel enables benchmarking and ranking of the different codes. In this paper, we construct KO codes, a computationaly efficient family of deep-learning driven (encoder, decoder) pairs that outperform the state-of-the-art reliability performance on the standardized AWGN channel. KO codes beat state-of-the-art Reed-Muller and Polar codes, under the low-complexity successive cancellation decoding, in the challenging short-to-medium block length regime on the AWGN channel. We show that the gains of KO codes are primarily due to the nonlinear mapping of information bits directly to transmit real symbols (bypassing modulation) and yet possess an efficient, high performance decoder. The key technical innovation that renders this possible is design of a novel family of neural architectures inspired by the computation tree of the bf Kronecker bf Operation (KO) central to Reed-Muller and Polar codes. These architectures pave way for the discovery of a much richer class of hitherto unexplored nonlinear algebraic structures. The code is available at hrefhttps://github.com/deepcomm/KOcodeshttps://github.com/deepcomm/KOcodes