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Concept# Codes polaires

Résumé

In information theory, a polar code is a linear block error-correcting code. The code construction is based on a multiple recursive concatenation of a short kernel code which transforms the physical channel into virtual outer channels. When the number of recursions becomes large, the virtual channels tend to either have high reliability or low reliability (in other words, they polarize or become sparse), and the data bits are allocated to the most reliable channels. It is the first code with an explicit construction to provably achieve the channel capacity for symmetric binary-input, discrete, memoryless channels (B-DMC) with polynomial dependence on the gap to capacity. Notably, polar codes have modest encoding and decoding complexity O(n log n), which renders them attractive for many applications. Moreover, the encoding and decoding energy complexity of generalized polar codes can reach the fundamental lower bounds for energy consumption of two dimensional circuitry to within an O(n^ε polylog n) factor for any ε > 0.
Polar codes have some limitations when used in industrial applications. Primarily, the original design of the polar codes achieves capacity when block sizes are asymptotically large with a successive cancellation decoder. However, with the block sizes used in industry, the performance of the successive cancellation is poor compared to well-defined and implemented coding schemes such as low-density parity-check code (LDPC) and turbo code. Polar performance can be improved with successive cancellation list decoding, but its usability in real applications is still questionable due to very poor implementation efficiencies caused by the iterative approach.
In October 2016, Huawei announced that it had achieved 27 Gbit/s in 5G field trial tests using polar codes for channel coding. The improvements have been introduced so that the channel performance has now almost closed the gap to the Shannon limit, which sets the bar for the maximum rate for a given bandwidth and a given noise level.

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Publications associées (50)

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This paper investigates the problem of secret key generation from correlated Gaussian random variables in the short blocklength regime. Short blocklengths are commonly employed in massively connected IoT sensor networks in 5G and beyond wireless systems. Polar codes have previously been shown to be applicable to the secret key generation problem, and are known to perform well for short blocklengths in the channel coding context. Inspired by these findings, we propose an explicit protocol based on polar codes for generating secret keys in the short blocklength regime. This protocol differs from previously proposed key generation protocols based on polar coding in two main ways: (i) we consider a Gaussian source for the key generation; (ii) we focus on the short blocklength regime. Simulation results show that the proposed protocol performs well even for very short blocklengths, especially if one can relax the BER/BLER requirements for the generated keys. They also demonstrate that the polar code based protocol outperforms a similar one using LDPC codes in place of polar codes, and that this advantage grows the shorter the blocklength becomes.

The beginning of 21st century provided us with many answers about how to reach the channel capacity. Polarization and spatial coupling are two techniques for achieving the capacity of binary memoryless symmetric channels under low-complexity decoding algorithms. Recent results prove that another way to achieve capacity is via symmetry, which is the case of the Reed-Muller and extended Bose-Chaudhuri-Hocquenghem (BCH) codes. However, this proof holds only for erasure channel and maximum a posteriori decoding, which is computationally intractable for the general channels.In the first part of this thesis, we talk about the performance improvements that an automorphism group of the code brings on board. We propose two decoding algorithms for the Reed-Muller codes, which are invariant under a large group of permutations and are expected to benefit the most. The former is based on plugging the codeword permutations in successive cancellation decoding, and the latter utilizes the code representation as the evaluations of Boolean monomials. However, despite the performance improvements, it is clear that the decoding complexity grows quickly and becomes impractical for moderate-length codes. In the second part of this thesis, we provide an explanation for this observation. We use the Boolean polynomial representation of the code in order to show that polar-like decoding of sufficiently symmetric codes asymptotically needs an exponential complexity. The automorphism groups of the Reed-Muller and eBCH codes limit the efficiency of their polar-like decoding for long codes, hence we either should focus on short lengths or find another way. We demonstrate that asymptotically same restrictions (although with a slower convergence) hold for more relaxed condition that we call partial symmetry. The developed framework also enables us to prove that the automorphism group of polar codes cannot include a large affine subgroup.In the last part of this thesis, we address a completely different problem. A device-independent quantum key distribution (DIQKD) aims to provide private communication between parties and has the security guarantees that come mostly from quantum physics, without making potentially unrealistic assumptions about the nature of the communication devices. After the quantum part of the DIQKD protocol, the parties share a secret key that is not perfectly correlated. In order to synchronize, some information needs to be revealed publicly, which makes this formulation equivalent to the asymmetric Slepian-Wolf problem that can be solved using binary linear error-correction codes. As any amount of the revealed information reduces the key secrecy, the utilized code should operate close to the finite-length limits. The channel in consideration is non-standard and, due to its experimental nature, it can actually slightly differ from the considered models. In order to solve this problem, we designed a simple scheme using universal SC-LDPC codes and used in the first successful experimental demonstration of DIQKD protocol.

Rüdiger Urbanke, Kirill Ivanov

The recently introduced polar codes constitute a breakthrough in coding theory due to their capacity-achieving property. This goes hand in hand with a quasilinear construction, encoding, and successive cancellation list decoding procedures based on the Plotkin construction. The decoding algorithm can be applied with slight modifications to Reed-Muller or eBCH codes, that both achieve the capacity of erasure channels, although the list size needed for good performance grows too fast to make the decoding practical even for moderate block lengths. The key ingredient for proving the capacity-achieving property of Reed-Muller and eBCH codes is their group of symmetries. It can be plugged into the concept of Plotkin decomposition to design various permutation decoding algorithms. Although such techniques allow to outperform the straightforward polar-like decoding, the complexity stays impractical. In this paper, we show that although invariance under a large automorphism group is valuable in a theoretical sense, it also ensures that the list size needed for good performance grows exponentially. We further establish the bounds that arise if we sacrifice some of the symmetries. Although the theoretical analysis of the list decoding algorithm remains an open problem, our result provides an insight into the factors that impact the decoding complexity.