Journal Article10.48550/arXiv.2303.13019
Construction Methods Based Minimum Weight Distribution for Polar Codes with Successive Cancellation List Decoding
TL;DR: In this paper , a greedy bit grouping reorder based MWD (BGR-MWD) algorithm was proposed to improve the performance of polar codes with SCL decoding, where the entropy constraint was introduced to establish a relationship between the list size and the ML performance.
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Abstract: In this paper, we focus on the construction methods based MWD for polar codes to improve the performance with successive cancellation list (SCL) decoding. We first propose an ordered and nested reliability sequence, namely MWD sequence, to improve the ML performance of polar codes and apply fast construction without the original channel information. In the MWD sequence, the synthetic channels are sorted by the partial MWD which is used to evaluate the influence of information bit on MWD and we prove the MWD sequence is the optimum sequence under ML decoding. Then, since the list size of SCL decoding is limited, we introduce an entropy constraint to establish a relationship between the list size and the ML performance and propose a heuristic and greedy construction method named bit grouping reorder based MWD (BGR-MWD) algorithm. In the algorithm, we divide the synthetic channels into groups by the partial MWD and greedily reorder the synthetic channels in some groups until the entropy constraint is satisfied. The simulation results show the MWD sequence is suitable for constructing polar codes with short code length. Meanwhile, the BGR-MWD algorithm has superior performance over the traditional construction methods for long code length.
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Figures
Fig. 3: Fig. 3(a) and Fig. 3(b) illustrate the minimum required SNRs of polar codes decoded by SCL decoding with L = 8 and L = 16 to achieve BLER ≤ 10−4 and BLER ≤ 10−3 under the AWGN channel with N = 128 and N = 256, respectively. 
Fig. 4: The BLER performance of polar codes constructed by different construction methods with N = 1024 and L = 32, where the information lengths of Fig. 4(a), Fig. 4(b) and Fig. 4(c) are K = 256, K = 512 and K = 768, respectively 
Fig. 2: The information sets obtained by BGR-MWD and BGR-MWD-PW algorithms with N = 32, K = 16, L = 2 and Eb/N0 = 1.25dB. 
TABLE I: The MWD sequence q128 with N = 128. 
Fig. 5: The BLER performance among different construction methods with different list sizes, where Fig. 5(a) and Fig. 5(b) show the comparison between BGR-MWD and GA algorithms and Fig. 5(c) and Fig. 5(d) provide the comparison between BGR-MWD-PW algorithm and PW sequence. 
TABLE II: The MWD sequence q256 with N = 256.
Citations
Performance Analysis for Polar Codes under Successive Cancellation List Decoding with Fixed List Size
TL;DR: In this article , the block error event of polar codes under successive cancellation list (SCL) decoding is composed of path loss (PL) and path selection (PS) error events, where the PL error event is that correct codeword is lost during the SCL decoding and the PS error event was that correct codeeword was reserved in the decoded list but not selected as decoded codewords, and the probability lower bound via the joint probability density of the log-likelihood ratios of information bits.
References
Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels
TL;DR: The paper proves that, given any B-DMC W with I(W) > 0 and any target rate R< I( W) there exists a sequence of polar codes {Cfrn;nges1} such that Cfrn has block-length N=2n, rate ges R, and probability of block error under successive cancellation decoding bounded as Pe(N,R) les O(N-1/4) independently of the code rate.
Channel Coding Rate in the Finite Blocklength Regime
TL;DR: It is shown analytically that the maximal rate achievable with error probability ¿ isclosely approximated by C - ¿(V/n) Q-1(¿) where C is the capacity, V is a characteristic of the channel referred to as channel dispersion, and Q is the complementary Gaussian cumulative distribution function.
List decoding of polar codes
Ido Tal,Alexander Vardy +1 more
- 03 Oct 2011
TL;DR: It appears that the proposed list decoder bridges the gap between successive-cancellation and maximum-likelihood decoding of polar codes, and devise an efficient, numerically stable, implementation taking only O(L · n log n) time and O( L · n) space.
How to Construct Polar Codes
Ido Tal,Alexander Vardy +1 more
TL;DR: A method for efficiently constructing polar codes is presented and analyzed, proving that for any fixed ε > 0 and all sufficiently large code lengths n, polar codes whose rate is within ε of channel capacity can be constructed in time and space that are both linear in n.
984
Efficient Design and Decoding of Polar Codes
TL;DR: It is shown that Gaussian approximation for density evolution enables one to accurately predict the performance of polar codes and concatenated codes based on them.
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