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IEEE Communications Letters

Polar-Code Construction of Golay Codes

Valerio Bioglio, Ingmar Land

Published Jan 15, 2018• 9 citations
Publisher

Abstract

This letter proposes a reinterpretation of the classical binary Golay codes as Polar codes with additional inner permutations and puncturing. This establishes a novel relationship between algebraic codes and codes based on Kronecker products, like polar codes. Naturally, this construction also allows for polar-code-type decoding of the Golay codes using successive cancellation list decoding. Using the Golay code as a driving example, we conjecture that other algebraic codes may be represented in a similar way.

DOI
10.1109/lcomm.2018.2793273

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This letter proposes a reinterpretation of the classical binary Golay codes as Polar codes with additional inner permutations and puncturing. This establishes a novel relationship between algebraic codes and codes based on Kronecker products, like polar codes. Naturally, this construction also allows for polar-code-type decoding of the Golay codes using successive cancellation list decoding. Using the Golay code as a driving example, we conjecture that other algebraic codes may be represented in a similar way.
This paper describes an adaptation of a polar code decoding technique in favor of the extended Golay code. Based on the bridge provided by a permutation matrix between the code words of these two classes of codes, the Golay code can be decoded by any polar code technique. Contrary to the successive-cancellation list technique which is characterized by a serial estimation of the bits, we propose in this work an adaptation of the simplified successive-cancellation list technique to polar codes equivalent to the Golay code. The simulations have achieved the performance of a maximum likelihood decoding, with the low decoding complexity of polar codes, compared to one of the universal decoders of linear codes most known in the literature.
A new permutation decoding approach for polar codes is presented. The proposed decoder can be used to achieve the error correction performance of a successive cancellation list (SCL) decoder, while it can either improve latency of the SCL decoder or decrease its hardware complexity. As opposed to the SCL algorithm, the proposed permutation algorithm can be implemented without sorting operation. It is shown that the error correction performance of the algorithm is similar to that of the SCL decoder for polar codes. Moreover, a new construction aiming to improve the error correction performance of polar codes under the proposed algorithm is presented.
Construction and Decoding Algorithms for for Polar Codes based on 2×22\times22×2 Non-Binary Kernels
Polar codes based on 2×22\times22×2 non-binary kernels are discussed in this work. The kernel over GF(q)\text{GF}(q)GF(q) is selected by maximizing the polarization effect and using Monte-Carlo simulation. Belief propagation (BP) and successive cancellation (SC) based decoding algorithms are extended to non-binary codes. Additionally, a successive cancellation list (SCL) decoding with a pruned tree is proposed. Simulation results show that the proposed decoder performs very close to a conventional SCL decoder with significantly lower complexity.
In this work, we propose a decoding method of Golay codes from the perspective of Polarization Adjusted Convolutional (PAC) codes. By invoking Forney's cubing construction of Golay codes and their generators G∗(8,7)/(8,4)G^*(8,7)/(8,4)G∗(8,7)/(8,4), we found different construction methods of Golay codes from PAC codes, which result in an efficient parallel list decoding algorithm with near-maximum likelihood performance. Compared with existing methods, our method can get rid of index permutation and codeword puncturing. Using the new decoding method, some related lattices, such as Leech lattice Λ24Λ_{24}Λ24​ and its principal sublattice H24H_{24}H24​, can be also decoded efficiently.
Polar coding has found its way into many realms in communications and information theory. In most implementation setups, they are accompanied with the list successive cancellation (LSC) decoding algorithm which is shown to provide a superior error performance compared to the original successive cancellation (SC) decoding method. While the SC decoding is fairly well-studied, the exact math behind LSC's superior performance still remains to be of mystery. Multiple techniques have been proposed to further improve the LSC's error performance or to reduce its computational complexity, which are usually motivated by heuristic reasons and shown through numerical simulations. Most notable example is the CRC-aided LSC, which drastically improves the LSC's performance by concatenating the polar code with some high-rate cyclic redundancy check (CRC) codes.In this paper, we present polar codes that are concatenated with an underlying high-rate convolutional code, which are shown to have superior performances over CRC-aided LSC. We also present a computationally-efficient decoding algorithm for these codes which resembles the techniques used in the Viterbi algorithm, and hence is called the convolutional decoding algorithm. To do this, we revisit the error analysis of the original SC decoding along with the concept of Arıkan's helper genie. We address some shortcomings of the CRC-aided LSC and discuss how to turn around them by emulating a convolutional code instead of a CRC code. Contrary to CRC codes, most of the convolutional codes are not a proper choice for concatenation with polar codes. We introduce the bucketing algorithm to construct suitable punctured convolutional codes for this purpose. The proposed framework can accommodate any such underlying convolutional code, which allows one to search for the optimal convolutional code based on their design parameters.