Polar codes are known as the first provable code construction to achieve Shannon capacity for arbitrary symmetric binary-input channels. Although, there exist efficient sub-optimal decoders with reduced complexity for polar codes, the complexity of the optimum ML decoder increases exponentially. Hence the optimum decoder is infeasible for the practical implementation of polar coding. In this paper, our motivation is about developing efficient ML decoder with reduced complexity. In this purpose, polar code based sphere decoding algorithm is proposed with the optimal performance. Additionally, proposed technique exploits two properties of polar coding to reduce decoding complexity. By this way, the reduced complexity of optimal decoding is only cubic, not exponential.
In this paper, we propose efficient maximumlikelihood (ML) decoding for binary Kronecker product-based (KPB) codes. This class of codes, have a matrix defined by the n-fold iterated Kronecker product Gn = F ⊗n of a binary upper-triangular kernel matrix F, where some columns are suppressed given a specific puncturing pattern. Polar and Reed-Muller codes are well known examples of such KPB codes.The triangular structure of Gn enables to perform ML decoding as a binary tree search for the closest codeword to the received point. We take advantage of the highly regular fractal structure of Gn and the "tree folding" technique to design an efficient ML decoder, enabling to decode relatively longer block lengths than with a standard binary tree search. The tree κfolding operation transforms the binary tree with N levels into a non-binary tree with N/2 κ levels, where the search can be significantly accelerated by a suitable ordering of the branch metrics. For a given κ we can find ( n κ ) different folding which lead to decoders with different complexity, for a given code.Using the proposed folded tree decoder, we provide exact ML performances of some Reed-Muller and polar codes over a binary AWGN channel for the block length up to 256.
Abstract-Polar codes are the first explicit class of codes that are provably capacity-achieving under the successive cancelation (SC) decoding. As a suboptimal decoder, SC has quasi-linear complexity N (1 + log N ) in the code length N . In this paper, we propose a new non-binary SC decoder with reduced complexity) based on the folding operation, which was first proposed in [11] to implement folded tree maximum-likelihood decoding of polar codes. Simulation results for the additive white Gaussian noise channel show that folded SC decoders can achieve the same error performance of standard SC by suitable selecting the folding of the polar code.
Abstract-Polar coding is known as the first provably capacityachieving coding scheme under low-complexity suboptimal successive cancelation decoding (SCD). The large error-correction capability of finite-length polar codes is mostly achieved with relatively long codes. SCD is the conventional decoder for polar codes and exhibits a quasi-linear complexity in terms of the code length. Practical decoder schemes with low latency are important for high-speed polar coding applications. In this letter, we propose a nonbinary multiple folded SCD scheme to reduce the decoding latency of standard binary polar codes. Multiple foldings were first proposed to improve the efficiency of folded tree maximumlikelihood decoder for Kronecker product-based codes. By successively applying the folding operation κ times on the SCD, for a code length N , the latency is reduced from 2N − 1 to (N/2 κ−1 ) − 1 time slots, assuming full parallelization. We show that multiple folded SCD can be effectively implemented for up to κ = 3 foldings due to memory limitations. This decoder achieves exactly the same performance of the original SCD with significantly reduced latency.
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