Polar Coding and Linear Decoding

• URL: http://arxiv.org/abs/2507.19695v1
• Date: Fri, 25 Jul 2025 22:23:14 GMT
• Title: Polar Coding and Linear Decoding
• Authors: Geraldo A. Barbosa,
• Abstract summary: Polar encoding, described by Arikan in IEEE Transactions on Information Theory, Vol. 55, No. 7, July 2009, was a milestone for telecommunications.<n>A Polar code distributes information among high and low-capacity channels, showing the possibility of achieving perfect channel capacity.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Polar encoding, described by Arikan in IEEE Transactions on Information Theory, Vol. 55, No. 7, July 2009, was a milestone for telecommunications. A Polar code distributes information among high and low-capacity channels, showing the possibility of achieving perfect channel capacity. The high-capacity channels allow almost noiseless transmission of data. When these channels are not high noise, reliability is achieved in the signal transmission. It starts to compete against codes such a Low-Density Parity-Check (LDPC) codes. Polar code can be also considered error correcting, based on the redundancy inherent in its structure. This feature makes polar encoding also applicable to digital quantum-resistant cryptography protocols. This work explores linear decoding at a first or single trial in the case of small losses or small number of bit-flipping, and repeated transmission for medium level losses. This is distinct from Arikans successive probabilistic decoding by application of probabilistic rules. Linear decoding is done directly from solving the linear equations connecting the codewords x and the received signals y after transmission via noisy channels. Numerical examples will be shown. Along with this work, programming in Mathematica language was used. Codes are available for copy-and- paste for Mathematica users to immediately try the described formalism.

Related papers

• Learning Linear Block Error Correction Codes [62.25533750469467]
  We propose for the first time a unified encoder-decoder training of binary linear block codes. We also propose a novel Transformer model in which the self-attention masking is performed in a differentiable fashion for the efficient backpropagation of the code gradient.
  arXiv  Detail & Related papers  (2024-05-07T06:47:12Z)
• Estimating the Decoding Failure Rate of Binary Regular Codes Using Iterative Decoding [84.0257274213152]
  We propose a new technique to provide accurate estimates of the DFR of a two-iterations (parallel) bit flipping decoder.<n>We validate our results, providing comparisons of the modeled and simulated weight of the syndrome, incorrectly-guessed error bit distribution at the end of the first iteration, and two-itcrypteration Decoding Failure Rates (DFR)
  arXiv  Detail & Related papers  (2024-01-30T11:40:24Z)
• Flexible polar encoding for information reconciliation in QKD [2.627883025193776]
  Quantum Key Distribution (QKD) enables two parties to establish a common secret key that is information-theoretically secure. Errors that are generally considered to be due to the adversary's tempering with the quantum channel need to be corrected using classical communication over a public channel. We show that the reliability sequence can be derived and used to design an encoder independent of the choice of decoder.
  arXiv  Detail & Related papers  (2023-11-30T16:01:10Z)
• Fault-Tolerant Quantum Memory using Low-Depth Random Circuit Codes [0.24578723416255752]
  Low-depth random circuit codes possess many desirable properties for quantum error correction. We design a fault-tolerant distillation protocol for preparing encoded states of one-dimensional random circuit codes. We show through numerical simulations that our protocol can correct erasure errors up to an error rate of $2%$.
  arXiv  Detail & Related papers  (2023-11-29T19:00:00Z)
• Fault-tolerant Coding for Entanglement-Assisted Communication [46.0607942851373]
  This paper studies the study of fault-tolerant channel coding for quantum channels. We use techniques from fault-tolerant quantum computing to establish coding theorems for sending classical and quantum information in this scenario. We extend these methods to the case of entanglement-assisted communication, in particular proving that the fault-tolerant capacity approaches the usual capacity when the gate error approaches zero.
  arXiv  Detail & Related papers  (2022-10-06T14:09:16Z)
• Denoising Diffusion Error Correction Codes [92.10654749898927]
  Recently, neural decoders have demonstrated their advantage over classical decoding techniques. Recent state-of-the-art neural decoders suffer from high complexity and lack the important iterative scheme characteristic of many legacy decoders. We propose to employ denoising diffusion models for the soft decoding of linear codes at arbitrary block lengths.
  arXiv  Detail & Related papers  (2022-09-16T11:00:50Z)
• Adversarial Neural Networks for Error Correcting Codes [76.70040964453638]
  We introduce a general framework to boost the performance and applicability of machine learning (ML) models. We propose to combine ML decoders with a competing discriminator network that tries to distinguish between codewords and noisy words. Our framework is game-theoretic, motivated by generative adversarial networks (GANs)
  arXiv  Detail & Related papers  (2021-12-21T19:14:44Z)
• Dense Coding with Locality Restriction for Decoder: Quantum Encoders vs. Super-Quantum Encoders [67.12391801199688]
  We investigate dense coding by imposing various locality restrictions to our decoder. In this task, the sender Alice and the receiver Bob share an entangled state.
  arXiv  Detail & Related papers  (2021-09-26T07:29:54Z)
• KO codes: Inventing Nonlinear Encoding and Decoding for Reliable Wireless Communication via Deep-learning [76.5589486928387]
  Landmark codes underpin reliable physical layer communication, e.g., Reed-Muller, BCH, Convolution, Turbo, LDPC and Polar codes. In this paper, we construct KO codes, a computationaly efficient family of deep-learning driven (encoder, decoder) pairs. KO codes beat state-of-the-art Reed-Muller and Polar codes, under the low-complexity successive cancellation decoding.
  arXiv  Detail & Related papers  (2021-08-29T21:08:30Z)
• Polar Codes for Quantum Reading [0.0]
  We show how to construct low complexity encoding schemes that are interesting for channel discrimination. We also show that the error probability of the scheme proposed decays exponentially with respect to the code length. An analysis of the optimal quantum states to be used as probes is given.
  arXiv  Detail & Related papers  (2020-12-14T01:24:11Z)
• Combining hard and soft decoders for hypergraph product codes [0.3326320568999944]
  Hypergraph product codes are constant-rate quantum low-density parity-check (LDPC) codes equipped with a linear-time decoder called small-set-flip (SSF) This decoder displays sub-optimal performance in practice and requires very large error correcting codes to be effective. We present new hybrid decoders that combine the belief propagation (BP) algorithm with the SSF decoder.
  arXiv  Detail & Related papers  (2020-04-23T14:48:05Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.