SCIENCE CHINA Information Sciences, Volume 62 , Issue 8 : 080303(2019) https://doi.org/10.1007/s11432-019-9905-7

## Partial CRC-aided decoding of 5G-NR short codes using reliability information

• AcceptedMay 23, 2019
• PublishedJul 11, 2019
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### Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant No. 61771133) and in part by National Science Technology Projects of China (Grant No. 2018ZX03001002).

### References

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• Figure 1

(Color online) The systematic structure of CRC-LDPC/polar codes. (a) CRC-LDPC codes; (b) CRC-polar codes.

• Table 1   Complexity comparison of different algorithms for polar codes
 Algorithm Equivalent addition numbers CASCL $\zeta_{s}=~L\cdot~N\cdot~\log_{2}{N}~+~K\cdot~L\cdot~\log_{2}{2L}~$ OSD$(i,P-\hat{P})$ $\zeta_{o}=\frac{1}{2}\hat{A}(\hat{A}-1)N$$+(N-\hat{A})(2\hat{A}-1)$$+\sum_{i=1}^{s}\binom{\hat{A}}{i}2(N-\hat{A})i$ Proposed $\zeta_{p}=\zeta_{s}+R_{f}\cdot~\zeta_{o}~$
•

Algorithm 1 OSD algorithm with partial CRC aided

Require:reliability information $~{\boldsymbol{R}}~$, input data of receiver $~{\boldsymbol{Y}}~$, union generator matrix $~{{\hat{\boldsymbol~G}}}_{I}~$;

Output: optimal codeword ${\boldsymbol{C}}_{\rm~op}$, CRC decision;

Sort the absolute value of $~{\boldsymbol{R}}~$ in descending order, get $~\pi_{1}({\boldsymbol{R}})~$;

Swap the corresponding column of $~{{\hat{\boldsymbol~G}}}_{I}~$, get $\pi_{1}({{\hat{\boldsymbol~G}}}_{I})$;

Do Gaussian elimination (GE) on $\pi_{1}({{\hat{\boldsymbol~G}}}_{I})$, make additional column swaps $\pi_{2}$ when there is all-zero column, get systematic matrix ${{\tilde{\boldsymbol~G}}}_{I}={\rm{GE}}(\pi_{2}(\pi_{1}({{\hat{\boldsymbol~G}}}_{I})))$;

Do hard decision on $~{\boldsymbol{R}}~$, get codeword ${\boldsymbol{C}}$;

Make corresponding two swaps on ${\boldsymbol{C}}$, get $\tilde{\boldsymbol{C}}=\pi_{2}(\pi_{1}(\boldsymbol{C}))$;

Do order-$i$ flipping on the basis of $\tilde{\boldsymbol{C}}$, get a code set ${\mathcal{C}_{f}}=\{{\boldsymbol{C}}_{f,j},~j=1,2,\ldots,\binom{\hat{A}}{i}\}$;

for $j=1,2,\ldots,\binom{\hat{A}}{i}$

${\boldsymbol{Y}}_{f,j}={\boldsymbol{C}}_{f,j}{{\tilde{\boldsymbol~G}}}_{I}$;

Calculate Euclidean distance between $~{\boldsymbol{Y}}_{f,j}~$ and $~{\boldsymbol{Y}}~$;

end for

Find $~{\boldsymbol{Y}}_{f,j}~$ with minimum Euclidean distance and the corresponding codeword $~{\boldsymbol{C}}_{f,j}~=~{\tilde{\boldsymbol{C}}}_{\rm~op}~$;

Recover the codeword in original sequence ${\boldsymbol{C}}_{\rm~op}=\pi_{1}^{-1}(\pi_{2}^{-1}({\tilde{\boldsymbol{C}}}_{\rm~op}))~$;

Do CRC testing on ${\boldsymbol{C}}_{\rm~op}$ and make a final decision if ${\boldsymbol{C}}_{\rm~op}$ is a correct codeword.

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