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1. Using Minors to Construct Generator Matrices for Quasi-Cyclic LDPC Codes

   https://doi.org/10.1109/ISIT50566.2022.9834862  

   Smarandache, Roxana; Gomez-Fonseca, Anthony; Mitchell, David G. (June 2022, 2022 IEEE International Symposium on Information Theory (ISIT))

   This paper gives a simple method to construct generator matrices with polynomial entries (and hence offers an alternative encoding method to the one commonly used) for all quasi-cyclic low-density parity-check (QC-LDPC) codes, even for those that are rank deficient. The approach is based on constructing a set of codewords with the desired total rank by using minors of the parity-check matrix. We exemplify the method on several well-known and standard codes. Moreover, we explore the connections between the minors of the parity-check matrix and the known upper bound on minimum distance and provide a method to compute the rank of any parity-check matrix representing a QC-LDPC code, and hence the dimension of the code, by using the minors of the corresponding polynomial parity-check matrix. 

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2. Nested LDPC Codes for Random Access Communication

   https://doi.org/10.1109/ISIT45174.2021.9518171  

   Liu, Yuxin; Effros, Michelle (July 2021, 2021 IEEE International Symposium on Information Theory (ISIT))

   This paper proposes a nested low-density parity-check (LDPC) code design. Combining this nested LDPC code with the random access coding strategy introduced by Yavas, Kostina, and Effros yields a random access LDPC (RA-LDPC) code for reliable communication in random access communication environments where neither the transmitters nor the receiver knows which or even how many transmitters wish to communicate at each moment. Coordination is achieved using sparse scheduled feedback. Bounds on the finite-blocklength performance of the RA-LDPC code under maximum likelihood (ML) decoding are derived using both error exponent and dispersion style analyses. Results include bounds on the penalty of the RA-LDPC code as a function of the LDPC code densities. 

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3. Window code parameters of spatially-coupled LDPC codes

   https://doi.org/10.1142/S0219498825501270  

   McMillon, Emily; Kelley, Christine A (June 2024, Journal of Algebra and Its Applications)

   In this paper, we define a window code to be the portion of a Spatially-coupled low-density parity check (SC-LDPC) code seen by a single iteration of a windowed decoder. We consider the design of SC-LDPC codes for windowed decoding via optimization of the window code. In particular, because iterative decoding is optimal on codes with cycle-free graph representations, we ask fundamental questions about the construction and parameters of cycle-free window codes. We show that it is possible to have an SC-LDPC code with cycles and with cycle-free window codes. We consider the relationship between the distance of the window code and the distance of the SC-LDPC code. Further, we show that SC-LDPC codes with MDS window codes exist, and all such codes are asymptotically bad. This work gives insight into the tradeoffs between window code parameters and performance of the SC-LDPC code. 

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4. Rate-Compatible, Bandwidth-Efficient, Low-Density Parity-Check (LDPC) Codes for Aeronautical Telemetry

   https://doi.org/10.3390/e26121045  

   Cummins, Andrew D; Mitchell, David G; Perrins, Erik (December 2024, Entropy)

   Low-density parity-check (LDPC) codes form part of the IRIG-106 standard and have been successfully deployed for the Telemetry Group version of shaped-offset quadrature phase shift keying (SOQPSK-TG) modulation. Recently, LDPC code solutions have been proposed and optimized for continuous phase modulations (CPMs), including pulse code modulation/frequency modulation (PCM/FM) and the multi-h CPM developed by the Advanced-Range TeleMetry program (ARTM CPM), the latter of which was shown to perform around one dB from channel capacity. In this paper, we consider the effect of the random puncturing and shortening of these LDPC codes to further improve spectrum efficiency. We perform asymptotic analyses of the ARTM0 code ensembles and present numerical simulation results that affirm the robust decoding performance promised by LDPC codes designed for ARTM CPM. 

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5. 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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