[Paper Review] Symbolic Stochastic Chase Decoding of Reed-Solomon and BCH Codes

Hossein Mani, Saied Hemati|arXiv (Cornell University)|Jul 14, 2017

Coding theory and cryptography3 citations

TL;DR

This paper proposes the Symbolic-Stochastic Chase Decoding Algorithm (S-SCA) for Reed-Solomon and BCH codes, which uses randomized, likelihood-based test-vector generation and a K-D tree to reduce complexity while approaching maximum-likelihood performance. The S-SCA achieves up to 2 dB gain over conventional Symbolic-Chase decoding with 1K iterations for (31,25) RS codes over 32-QAM, and over 5 dB gain in Rayleigh fading, with near-ML performance at low complexity due to efficient search in high-likelihood symbol regions.

ABSTRACT

This paper proposes the Symbolic-Stochastic Chase Decoding Algorithm (S-SCA) for the Reed-Solomon (RS) and BCH codes. By efficient usage of void space between constellation points for $q$-ary modulations and using soft information at the input of the decoder, the S-SCA is capable of outperforming conventional Symbolic-Chase algorithm (S-CA) with less computational cost. Since the S-SCA starts with the randomized generation of likely test-vectors, it reduces the complexity to polynomial order and also it does not need to find the least reliable symbols to generate test-vectors. Our simulation results show that by increasing the number of test-vectors, the performance of the algorithm can approach the ML bound. The S-SCA($1K$) provides near $2$ dB gain in comparison with S-CA($1K$) for $(31, 25)$ RS code using $32$-QAM. Furthermore, the algorithm provides near $3$ dB further gain with $1K$ iteration compared with S-CA($65K$) when $(255, 239)$ RS code is used in an AWGN channel. For the Rayleigh fading channel and the same code, the algorithm provides more that $5$ dB gain. Also for $(63, 57)$ BCH codes and $8$-PSK modulation the proposed algorithm provides $3$dB gain with less complexity. This decoder is Soft-Input Soft-Output (SISO) decoder and is highly attractive in low power applications. Finally, the Symbolic-Search Bitwise-Transmission Stochastic Chase Algorithm (SSBT-SCA) was introduced for RS codes over BPSK transmission that is capable of generating symbolic test-vectors that reduce complexity and mitigate burst errors.

Motivation & Objective

• To develop a soft-input soft-output (SISO) decoding algorithm for Reed-Solomon and BCH codes that approaches maximum-likelihood (ML) performance with lower computational complexity than existing symbolic-chase methods.
• To overcome the exponential complexity of traditional symbolic-chase decoding by replacing deterministic, reliability-based test-vector generation with a randomized, likelihood-driven approach.
• To reduce decoding complexity by using a K-D tree to efficiently search for unique, high-likelihood test-vectors in the $q$-ary symbol space.
• To enable low-power, high-performance decoding in applications such as satellite communications and data storage by achieving near-ML performance with polynomial-time complexity.
• To demonstrate that increasing the number of test-vectors via stochastic sampling leads to a smooth convergence toward the ML bound, especially in fading channels.

Proposed method

• The S-SCA generates test-vectors stochastically by sampling symbols based on their likelihood, avoiding the need to identify the least reliable symbols as in conventional symbolic-chase decoding.
• It employs a K-D tree data structure to index and efficiently search for unique test-vectors in the $q$-ary codeword space, reducing the search complexity to $\mathcal{O}(n^2 N \log N)$, where $N$ is the number of test-vectors.
• The algorithm uses soft information from the channel input to guide the random generation of test-vectors, focusing on regions of high likelihood in the constellation space.
• By leveraging the void space between constellation points in $q$-ary modulations, the S-SCA improves error correction performance without increasing modulation density.
• The decoder is designed as a soft-input soft-output (SISO) block, enabling integration into concatenated coding systems for improved overall performance.
• The S-SCA is extended to a bitwise transmission variant, SSBT-SCA, for BPSK-modulated RS codes, which further reduces complexity and improves burst error resilience.

Experimental results

Research questions

• RQ1 Can a stochastic, likelihood-based test-vector generation method reduce the complexity of symbolic-chase decoding while maintaining or improving error correction performance?
• RQ2 How does the performance of the S-SCA compare to the conventional Symbolic-Chase Algorithm (S-CA) in terms of frame error rate (FER) and SNR gain across different code rates and modulation schemes?
• RQ3 To what extent can the S-SCA approach the maximum-likelihood (ML) bound, and how does the number of iterations affect convergence speed and performance gain?
• RQ4 What is the trade-off between the number of test-vectors and the resulting complexity versus FER performance in AWGN and Rayleigh fading channels?
• RQ5 Can the K-D tree indexing technique effectively reduce the search complexity of test-vectors in high-dimensional $q$-ary spaces without sacrificing performance?

Key findings

• For the (31,25) RS code with 32-QAM, S-SCA(1K) achieves a 2 dB gain over S-CA(1K), demonstrating significant performance improvement with lower complexity.
• For the (255,239) RS code in an AWGN channel, S-SCA(1K) provides more than 3 dB gain over S-CA(65K), showing superior convergence to the ML bound.
• In a Rayleigh fading channel, the S-SCA provides over 5 dB gain compared to S-CA(65K), highlighting its robustness in fading environments.
• For the (63,57) BCH code with 8-PSK modulation, the S-SCA achieves a 3 dB gain over conventional methods with reduced complexity.
• At FER = $10^{-3}$, the S-SCA(1K) for (31,25) RS code achieves a gap of only 0.41 dB to the ML bound, improving to 0.043 dB at 1M iterations.
• The performance gain diminishes beyond a certain iteration threshold: increasing from 512K to 1M iterations yields only a 0.03 dB improvement, indicating diminishing returns.