Towards Optimal Decoding for Polar Codes

In the conventional successive cancellation (SC) decoder for polar codes, all the future bits to be estimated later are treated as random variables. However, polar codes inevitably involve frozen bits, and their concatenated coding schemes also include parity bits causally generated from the past bits estimated earlier. We refer to the frozen and parity bits located behind a target decoding bit as its future constraints (FCs). Although the values of FCs are deterministic given the past estimates, they have not been exploited in the conventional SC-based decoders, not leading to optimality. In this paper, we propose SC-check (SCC) and belief-propagation SCC (BP-SCC) decoding algorithms in order to leverage FCs in decoding.We further devise a tree search technique based on stack-based backjumping (SBJ) to solve dynamic constraint satisfaction problems (CSPs) formulated by FCs. Over the binary erasure channel (BEC), numerical results show that a combination of the BP-SCC algorithm and the SBJ tree search technique achieves the erasure recovery performance close to the dependence testing (DT) bound, a bound of achievable finite-length performance

Error-Aware SCFlip Decoding of Polar Codes

The successive-cancellation flip (SCFlip) decoder and its variants provide a significant coding gain with the average complexity practically identical to that of the successive cancellation (SC) decoder in a wide range of signal-to-noise ratios (SNRs). But, they suffer from high complexity and long latency when the SNR decreases, since the average number of extra decoding attempts becomes inevitably large. To mitigate this problem, we propose a novel SCFlip decoder, called an error-aware SCFlip (EA-SCFlip) decoder, for distributed cyclic-redundancy-check (CRC) polar codes. Based on the distributed CRC bits, it employs early termination at each extra decoding attempt so that it can reduce the decoding complexity and latency on the average. It also reduces the search space of candidate bit-flips in the dynamic building of the bit-flip list by exploiting the parity-check relationship (PCR) of the first error-detected CRC bit at each extra decoding attempt. Furthermore, we propose a greedy algorithm to design a distributed CRC code such that the obtained PCRs make the early-error-detection capability of the EA-SCFlip decoder as high as possible. Numerical results demonstrate that the EA-SCFlip decoder can indeed achieve an early termination gain as well as a complexity reduction, when a polar code is concatenated with the distributed CRC code designed by the proposed algorithm.11Ysciescopu

부분적 CRC-polar 부호의 설계 방법

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A Two-Bit Precision Decoding Algorithm for LDPC Codes Over the Binary Symmetric Channel

The min-sum algorithm as an approximation of the sum-product algorithm (SPA) for decoding of LDPC codes has much lower computational complexity, but slightly worse performance than the SPA. However, severe performance degradation might occur when it is realized in hardware due to the finite precision effect, especially in two-bit precision. In order to overcome this problem, we propose a two-hit precision decoding algorithm. The proposed algorithm employs one of several decoders adaptively, based on the syndrome weight of the estimated decoder output every iteration. Numerical results show that it has similar performance to the SPA over the binary symmetric channel.11Nsciescopu

비이진 LDPC 부호를 위한 개선된 수렴속도를 가지는 다중투표 심볼반전 복호 알고리즘

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A new class of balanced near-perfect nonlinear mappings and its application to sequence design

A mapping from BBZN to BBZM can be directly applied for the design of a sequence of period N with alphabet size M, where BBZ N denotes the ring of integers modulo N. The nonlinearity of such a mapping is closely related to the autocorrelation of the corresponding sequence. When M is a divisor of N, the sequence corresponding to a perfect nonlinear mapping has perfect autocorrelation, but it is not balanced. In this paper, we study balanced near-perfect nonlinear (NPN) mappings applicable for the design of sequence sets with low correlation. We first construct a new class of balanced NPN mappings from BBZ-{p{2}-p} to BBZp for an odd prime p. We then present a general method to construct a frequency-hopping sequence (FHS) set from a nonlinear mapping. By applying it to the new class, we obtain a new optimal FHS set of period p{2}-p with respect to the Peng-Fan bound, whose FHSs are balanced and optimal with respect to the Lempel-Greenberger bound. Moreover, we construct a low-correlation sequence set with size p, period p^{2}-p, and maximum correlation magnitude p from the new class of balanced NPN mappings, which is asymptotically optimal with respect to the Welch bound.clos

Algebraic Construction of Structurally Shaped Polar Codes

Shaped polar codes were recently proposed to improve the spectral efficiency of polar coded modulation, based on probabilistic shaping. The key principle for them is to adjust the probability of occurrence of 1's in each codeword by introducing the shaping bits, which makes the distribution of modulated symbols close to the Gaussian distribution. But, they have a serious problem from a practical viewpoint, since additional polar decoding is required to determine the shaping bits. In this paper, we investigate an algebraic approach to the design of structurally shaped polar codes. We first select a shaping set in an algebraic and systematic way and determine the shaping bits by using simple binary and integer operations without any additional polar decoding. We then propose a method to construct structurally shaped polar codes, and analyze their shaped probability in an approximate way. Finally, we present the encoding and decoding procedures for the proposed shaped polar codes. Numerical results show that the proposed shaping scheme has similar performance than the conventional shaping scheme regardless of whether cyclic redundancy check is employed, while the former can be implemented in an extremely simpler way than the latter.11Nsciescopu

동적 동결 비트를 사용한 극 부호 설계 방법

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