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1. ELF Codes: Concatenated Codes with an Expurgating Linear Function as the Outer Code

   https://doi.org/10.1109/ISTC57237.2023.10273535  

   Wesel, R.; Antonini, A.; Wang, L.; Sui, W.; Towell, B.; Grissett, H. (September 2023, IEEE)

   An expurgating linear function (ELF) is an outer code that disallows low-weight codewords of the inner code. ELFs can be designed either to maximize the minimum distance or to minimize the codeword error rate (CER) of the expurgated code. A list-decoding sieve can efficiently identify ELFs that maximize the minimum distance of the expurgated code. For convolutional inner codes, this paper provides analytical distance spectrum union (DSU) bounds on the CER of the concatenated code. For short codeword lengths, ELFs transform a good inner code into a great concatenated code. For a constant message size of K = 64 bits or constant codeword blocklength of N = 152 bits, an ELF can reduce the gap at CER 10−6 between the DSU and the random-coding union (RCU) bounds from over 1 dB for the inner code alone to 0.23 dB for the concatenated code. The DSU bounds can also characterize puncturing that mitigates the rate overhead of the ELF while maintaining the DSU-to-RCU gap. List Viterbi decoding guided by the ELF achieves maximum likelihood (ML) decoding of the concatenated code with a sufficiently large list size. The rate-K/(K+m) ELF outer code reduces rate and list decoding increases decoder complexity. As SNR increases, the average list size converges to 1 and average complexity is similar to Viterbi decoding on the trellis of the inner code. For rare large-magnitude noise events, which occur less often than the FER of the inner code, a deep search in the list finds the ML codeword. 

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2. On Sparse Regression LDPC Codes

   https://doi.org/10.1109/ISIT54713.2023.10206818  

   Ebert, Jamison R; Chamberland, Jean-Francois; Narayanan, Krishna R (August 2023, IEEE)

   Iterative decoding of graph-based codes and sparse recovery through approximate message passing (AMP) are two research areas that have seen monumental progress in recent decades. Inspired by these advances, this article introduces sparse regression LDPC codes (SR-LDPC codes) and their decoding. Sparse regression codes (SPARCs) are a class of error correcting codes that build on ideas from compressed sensing and can be decoded using AMP. In certain settings, SPARCs are known to achieve capacity; yet, their performance suffers at finite block lengths. Likewise, low-density parity-check (LDPC) codes can be decoded efficiently using belief propagation and can also be capacity achieving. This article introduces a novel concatenated coding structure that combines an LDPC outer code with a SPARC-inspired inner code. Efficient decoding for such a code can be achieved using AMP with a denoiser that performs belief propagation on the factor graph of the outer LDPC code. The proposed framework exhibits performance improvements over SPARCs and standard LDPC codes for finite block lengths and results in a steep waterfall in error performance, a phenomenon not observed in uncoded SPARCs. 

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3. Sparse Regression LDPC Codes

   https://doi.org/10.1109/TIT.2024.3496486  

   Ebert, Jamison R; Chamberland, Jean-Francois; Narayanan, Krishna R (January 2025, IEEE Transactions on Information Theory)

   This article introduces a novel concatenated coding scheme called sparse regression LDPC (SR-LDPC) codes. An SR-LDPC code consists of an outer non-binary LDPC code and an inner sparse regression code (SPARC), whose respective field size and section sizes are equal. For such codes, an efficient decoding algorithm is proposed based on approximate message passing (AMP) that dynamically shares soft information between inner and outer decoders. This dynamic exchange of information is facilitated by a denoiser that runs belief propagation (BP) on the factor graph of the outer LDPC code within each AMP iteration. It is shown that this BP denoiser falls within the framework of non-separable denoising functions and subsequently, that state evolution holds for the proposed AMP-BP algorithm. Leveraging the rich structure of SR-LDPC codes, this article proposes an efficient low-dimensional approximate state evolution recursion that can be used for efficient hyperparameter tuning, thus paving the way for future work on optimal code design. Finally, numerical simulations demonstrate that SR-LDPC codes outperform contemporary codes over the AWGN channel for parameters of practical interest. SR-LDPC codes are shown to be viable means for obtaining shaping gains over the AWGN channel. 

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4. Concatenated Spatially Coupled LDPC Codes with Sliding Window Decoding for Joint Source-Channel Coding

   https://doi.org/10.1109/tcomm.2021.3126750  

   Golmohammadi, Ahmad; Mitchell, David G. (November 2021, IEEE Transactions on Communications)

   In this paper, a method for joint source-channel coding (JSCC) based on concatenated spatially coupled low-density parity-check (SC-LDPC) codes is investigated. A construction consisting of two SC-LDPC codes is proposed: one for source coding and the other for channel coding, with a joint belief propagation-based decoder. Also, a novel windowed decoding (WD) scheme is presented with significantly reduced latency and complexity requirements. The asymptotic behavior for various graph node degrees is analyzed using a protograph-based Extrinsic Information Transfer (EXIT) chart analysis for both LDPC block codes with block decoding and for SC-LDPC codes with the WD scheme, showing robust performance for concatenated SC-LDPC codes. Simulation results show a notable performance improvement compared to existing state-of-the-art JSCC schemes based on LDPC codes with comparable latency and complexity constraints. 

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5. Parallel Trellis-Stage-Combining BCJR for High-Throughput CUDA Decoder of CCSDS SCPPM

   Antonini, Amaael; Glukhov, Egor; Wesel, Richard; Divsalar, Dariush; Hamkins, Jon (March 2025, IEEE)

   The Consultative Committee for Space Data Systems (CCSDS) standard for high photon efficiency uses a serially-concatenated (SC) code to encode pulse position modulated laser light. A convolutional encoder serves as the outer code and an accumulator serves as the inner code. These two component codes are connected through an interleaver. This coding scheme is called Serially Concatenated convolutionally coded Pulse Position Modulation (SCPPM) and it is used for NASA's Deep Space Optical Communications (DSOC) experiment. For traditional decoding that traverses the trellis forwards and backwards according to the Bahl Cocke Jelinek and Raviv (BCJR) algorithm, the latency is on the order of the length of the trellis, which has 10,080 stages for the rate 2/3 DSOC code. This paper presents a novel alternative approach that simultaneously processes all trellis stages, successively combining pairs of stages into a meta-stage. This approach has latency that is on the order of the log base-2 of the number of stages. The new decoder is implemented using the Compute Unified Device Architecture (CUDA) platform on an Nvidia Graphics Processing Unit (GPU). Compared to Field Programmable Gate Array (FPGA) implementations, the GPU implementation offers easier development, scalability, and portability across GPU models. The GPU implementation provides a dramatic increase in speed that facilitates more thorough simulation as well as enables a shift from FPGA to GPU processors for DSOC ground stations. 

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