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Low-Complexity Decoding for Low-Rate Block Codes of Short Length Based on Concatenated Coding Structure

Mao-Chao Lin, Shih-Kai Lee, Pin Lin, Ching-Chang Lin, Chia-Chun Chen, Teng-Yuan Syu, Huang-Chang Lee

TL;DR

The work addresses decoding short, low-rate block codes with stringent latency by introducing concatenated coding to yield a more reliable MRIP frame via inner SISO outputs. By combining a Reed-Solomon outer code with binary inner codes and an enhanced MRIP, the approach enables A$^*$ decoding (and related OSD variants) to operate with smaller effective search spaces and lower complexity. Empirical results for $(n,k)=(128,36)$ show concatenated schemes—especially using a $(2,1,6)$ inner convolutional code—achieve BLERs near ML bounds with substantial complexity reductions compared to the baseline $(128,36)$ eBCH code; similar gains are demonstrated for nearby lengths and rates. The findings indicate that concatenated coding with improved MRIP can significantly improve short, low-rate code performance and decoding efficiency, with potential application to other short-length regimes as well.

Abstract

To decode a short linear block code, ordered statics decoding (OSD) and/or the $A^*$ decoding are usually considered. Either OSD or the $A^*$ decoding utilizes the magnitudes of the received symbols to establish the most reliable and independent positions (MRIP) frame. A restricted searched space can be employed to achieve near-optimum decoding with reduced decoding complexity. For a low-rate code with large minimum distance, the restricted search space is still very huge. We propose to use concatenated coding to further restrict the search space by proposing an improved MRIP frame. The improved MRIP frame is founded according to magnitudes of log likelihood ratios (LLRs) obtained by the soft-in soft-out (SISO) decoder for the inner code. We focus on the construction and decoding of several $(n,k)$ = (128,36) binary linear block codes based on concatenated coding. We use the (128,36) extended BCH (eBCH) code as a benchmark for comparison. Simulation shows that there exist constructed concatenated codes which are much more efficient than the (128,36) eBCH code. Some other codes of length 128 or close to 128 are also constructed to demonstrate the efficiency of the proposed scheme.

Low-Complexity Decoding for Low-Rate Block Codes of Short Length Based on Concatenated Coding Structure

TL;DR

The work addresses decoding short, low-rate block codes with stringent latency by introducing concatenated coding to yield a more reliable MRIP frame via inner SISO outputs. By combining a Reed-Solomon outer code with binary inner codes and an enhanced MRIP, the approach enables A decoding (and related OSD variants) to operate with smaller effective search spaces and lower complexity. Empirical results for show concatenated schemes—especially using a inner convolutional code—achieve BLERs near ML bounds with substantial complexity reductions compared to the baseline eBCH code; similar gains are demonstrated for nearby lengths and rates. The findings indicate that concatenated coding with improved MRIP can significantly improve short, low-rate code performance and decoding efficiency, with potential application to other short-length regimes as well.

Abstract

To decode a short linear block code, ordered statics decoding (OSD) and/or the decoding are usually considered. Either OSD or the decoding utilizes the magnitudes of the received symbols to establish the most reliable and independent positions (MRIP) frame. A restricted searched space can be employed to achieve near-optimum decoding with reduced decoding complexity. For a low-rate code with large minimum distance, the restricted search space is still very huge. We propose to use concatenated coding to further restrict the search space by proposing an improved MRIP frame. The improved MRIP frame is founded according to magnitudes of log likelihood ratios (LLRs) obtained by the soft-in soft-out (SISO) decoder for the inner code. We focus on the construction and decoding of several = (128,36) binary linear block codes based on concatenated coding. We use the (128,36) extended BCH (eBCH) code as a benchmark for comparison. Simulation shows that there exist constructed concatenated codes which are much more efficient than the (128,36) eBCH code. Some other codes of length 128 or close to 128 are also constructed to demonstrate the efficiency of the proposed scheme.
Paper Structure (27 sections, 1 theorem, 13 equations, 9 figures)

This paper contains 27 sections, 1 theorem, 13 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Review for the Basic OSD and $A^*$ algorithm
  3. MRIP
  4. OSD with order $\lambda$
  5. $A^*$ decoding
  6. PC-$\lambda$ and PC-out-$\lambda$
  7. Stopping Criterion
  8. Stack Design for $A^*$ Decoding
  9. Conventional Stack in Ascending Order
  10. Modified Stack without Ordering
  11. Performances Regarding Different Stack Designs and Thresholds
  12. Short Codes Based on Concatenation
  13. Possible Inner Codes
  14. Complexity of SISO decoding
  15. Improved MRIP Frame
  16. ...and 12 more sections

Key Result

Theorem 1

For the operation of the modified stack, a path $\hat{\bf c}$ which represents a goal node with $d_{H}(\hat{\bf c}_{0}^{k-1},\hat{\bf z}_{0}^{k-1})$ = $i$ will be searched before any path $\hat{\bf c}'$ which represents a goal node with $d_{H}(\hat{\bf c'}_{0}^{k-1},\hat{\bf z}_{0}^{k-1})$ = $i+1$ e

Figures (9)

  • Figure 1: BLER performances for (128,36) eBCH code using PC-out-$\lambda$ with $\lambda$=4 based on modified stack with size of 60000 and conventional stack with size of 30000 respectively and using stopping criteria based on $\alpha$ = 0, $\alpha$ = 0.05 and $M_{TH,\hat{c}}$ respectively.
  • Figure 2: Number of real-number operations for (128,36) eBCH code using PC-out-$\lambda$ with $\lambda$=4 based on modified stack with size of 60000 and conventional stack with size of 30000 respectively and using stopping criteria based on $\alpha$ = 0, $\alpha$ = 0.05 and $M_{TH,\hat{c}}$ respectively.
  • Figure 3: Variances $\sigma_{L}^2$ of LLR obtained from SISO decoders for various inner codes.
  • Figure 4: Means of LLR values in decreasing order for various inner codes. $E_s/N_0 = 3$ dB
  • Figure 5: Probability regarding the number of errors at MRIP for (128,36) codes, $E_b/N_0$ = 3 dB.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Theorem 1