SCIENCE CHINA Information Sciences, Volume 63 , Issue 10 : 202303(2020) https://doi.org/10.1007/s11432-019-2924-6

## Efficient stochastic successive cancellation listdecoder for polar codes

Xiao LIANG 1,2,3,4, Huizheng WANG 1,2,3,4, Yifei SHEN 1,2,3,4, Zaichen ZHANG 1,2,3,4, Xiaohu YOU 1,2,3,4, Chuan ZHANG 1,2,3,4,*
• ReceivedDec 29, 2019
• AcceptedMay 22, 2020
• PublishedSep 21, 2020
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### Abstract

Polar codes are one of the most favorable capacity-achieving codes owing to their simple structures and low decoding complexity. Successive cancellation list (SCL) decoders with large list sizes achieve performances very close to those of maximum-likelihood (ML) decoders. However, hardware cost is a severe problem because an SCL decoder with list size $L$ consists of $L$ copies of a successive cancellation (SC) decoder. To address this issue, a stochastic SCL (SSCL) polar decoder is proposed. Although stochastic computing can achieve a good hardware reduction compared with the deterministic one, its straightforward application to an SCL decoder is not well-suited owing to the precision loss and severe latency. Therefore, a doubling probability approach and adaptive distributed sorting (DS) are introduced. A corresponding hardware architecture is also developed. Field programmable gate array (FPGA) results demonstrate that the proposed stochastic SCL polar decoder can achieve a good performance and complexity tradeoff.

### Acknowledgment

This work was supported in part by National Key RD Program of China (Grant No. 2020YFB220-5503), National Natural Science Foundation of China (Grant Nos. 61871115, 61501116), Jiangsu Provincial NSF for Excellent Young Scholars (Grant No. BK20180059), Six Talent Peak Program of Jiangsu Province (Grant No. 2018-DZXX-001), Distinguished Perfection Professorship of Southeast University, Fundamental Research Funds for the Central Universities, and Student Research Training Program of Southeast University.

### References

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• Figure 5

(Color online) Candidate paths' selection in polar SCL decoder ($L=4$).

• Figure 8

(Color online) Binary-tree of stochastic SCL decoder in double-level ($L=2$).

• Figure 9

(Color online) Candidate paths' selection in polar SCL decoder with double-level ($L=4$).

• Figure 16

(Color online) Mixed node II for the last stage, also mixed node III for the first stage in SCL decoding.

• Figure 17

(Color online) The $8$-bit stochastic SC decoder with double-level decoding.

• Figure 18

(Color online) The $N$-bit stochastic SCL decoder with double-level decoding.

• Figure 19

(Color online) The architecture of list core module with ADS method and enlarging probability approach.

• Figure 20

(Color online) Decoding schedule for $N$-bit stochastic SCL decoder with double-level decoding.

•

Algorithm 1 Proposed ADS algorithm

setstretch0.7

Require:$~$ $s1:jk=\{00,01\}$; $s2:jk=\{00,10\}$; $s3:jk=\{00,01,10,11\}$; $\mathbf{I}$: $L$ previous path numbers; $[\mathbf{PPM}_{\rm~FC},~\mathbf{JK}]=\max\nolimits_{s1,~s2,~s3~}(\mathbf{PPM}(\hat{u}_{1}^{2i-2}|\hat{u}_{1}^{2i-1}=j,\hat{u}_{1}^{2i}=k))$; $\mathbf{PPM}_{\rm~NC}$, expanded paths list $\mathbb{P}$.

Output:$~$ Updated paths list $\mathbb{P}_{\mathrm{new}}=[\mathbb{P}(\mathbf{I}),~\mathbf{J},~\mathbf{K}]$ and PPM list $\mathbf{PPM}_{s}$.

$[\mathbf{PPM}_{s},~\mathbf{Ix},~\mathbf{JKx}]=\mathrm{sort}(\mathbf{PPM}_{\rm~FC},~{\rm~ascend})$;

$[\mathbf{PPM}_{1},~\mathbf{Iy},~\mathbf{JKy}]=\mathrm{sort}(\mathbf{PPM}_{\rm~NC},~{\rm~descend})$;

for $i=1:L-1$

$f=0$;

if $(\mathrm{PPM}_{1,i}\leqslant\mathrm{PPM}_{s,i})$ then

$f=1$;

else

$f=0$;

end if

if ($f==0$) then

Update $\mathbf{I}$: replace $\mathbf{I}\big(Ix(i)\big)$ with $\mathbf{I}\big(Iy(i)\big)$;

Update ${\boldsymbol~u}_{2i-1}{\boldsymbol~u}_{2i}$: $\mathbf{JK}\big(JKx(i)\big)=\mathbf{JK}\big(JKy(i)\big)$;

else

break;

end if

end for

Updated paths list $\mathbb{P}_{\mathrm{new}}=[\mathbb{P}(\mathbf{I}),~\mathbf{J},~\mathbf{K}]$ and PPM list $\mathbf{PPM}_{s}$.

• Table 1

Table 1The complexity of different modules of stochastic SCL decoder$^{\rm~a)}$

 Modules Mixed node I Mixed node II Mixed node III SC decoder$^{\rm~a)}$ SCL decoder$^{\rm~b)}$ XOR $1$ $1$ $1$ $N-1$ $NL/2-L+N/2$ NOT 3 6 5 $3N$ ${3NL}/{2}+{5N}/{2}$ AND $2$ $8$ $4$ $2N+4$ $NL+4L+2N$ 1-bit register $1$ $1$ $1$ $N-1$ ${NL}/{2}-L+{N}/{2}$ JK-FF $1$ $2$ $2$ $N$ ${NL}/{2}+N$ MUX $2$ – – $2(N-2)$ $NL+N-4L$ Counter – – – $4$ $4L$

a) $N$ and $L$ denote the code length and list size, respectively; b) Double-level decoding.

• Table 2

Table 2Comparison of the implementation results of several FPGA-based SCL architectures

 Decoder [20] [22] SSCL$^{\rm~a)}$ SSCL$^{\rm~a)}$ with $1$-level decoding with $2$-level decoding FPGA device$^{\rm~b)}$ Altera stratix V Xilinx Kintex 7 Altera stratix V Altera stratix V $(N,~L,~R)$ $(1024,~4,~0.5)$ $(1024,~4,~0.5)$ $(1024,~4,~0.5)$ $(1024,~4,~0.5)$ ALMs(A)/LUTs(X)$^{\rm~c)}$ $101160$ $142961$ $8080$ $8146$ Registers $13544$ $19795$ $2824$ $2862$ $f_{op}$ (MHz) – $42.66$ $462.9$ $445.2$ Latency (cycles) $4064$ $371$ $2^{20}$ $2^{19}$ TP (Mbps) – $115$ $0.452$ $0.871$

a) Stream length $l=1024$. b) All the FPGAs are manufactured on 28 nm process technology. c) An ALM on Altera FPGA can be used as a 6-input LUT.

•

Algorithm 2 ADS algorithm for $2^p$ level decoder

setstretch0.7

Require:$~$ $2^p$-level scheme, list size is $L$;

Output:$~$ Final $L$ candidates.

for $i=1:L$

Choose the best PPM as FC$_i$, the rest as $\mathbf{NC}_i=[{\rm~NC}_{i,1},~{\rm~NC}_{i,2},\ldots,~{\rm~NC}_{i,(2^{2^P}-1)L}]$;

Set $\mathbf{FC}=[{\rm~FC}_1,~{\rm~FC}_2,\ldots,~{\rm~FC}_L]$;

end for

Set candidates$=[{\rm~FC}_1,{\rm~FC}_2,\ldots,{\rm~FC}_L]$;

for $i=1:L$

Choose the best PPM of each $\mathbf{NC}_i$;

$\mathbf{NC^k}=[{\rm~NC}_{1k},~{\rm~NC}_{2k},\ldots,~{\rm~NC}_{Lk}]$;

end for

Select the worst PPM ${\rm~FC}_k$ of $\mathbf{FC}$;

Select the best PPM ${\rm~NC}_{pk}$ of $\mathbf{NC}^k$, $p$ denotes ${\rm~NC}_{pk}$ is from $\mathbf{NC}_{p}$;

if ${\rm~NC}_{pk}<{\rm~FC}_k$ then

Jump to line $21$;

else

Keep on;

end if

Replace ${\rm~FC}_k$ with ${\rm~NC}_{pk}$ in $\mathbf{candidates}$;

Cancel ${\rm~FC}_k$ from $\mathbf{FC}$, cancel ${\rm~NC}_{pk}$ from $\mathbf{NC}_p$;

Select the best PPM of $\mathbf{NC}_p$ as ${\rm~NC}_{pk}$;

Jump to line $10$;

Output final $L$ $\mathbf{candidates}$.

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