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1. Probabilistic Shaping for Trellis-Coded Modulation with CRC-Aided List Decoding

   https://doi.org/10.1109/TCOMM.2023.3237263  

   Wang, Linfang ; Song, Dan ; Areces, Felipe ; Wiegart, Thomas ; Wesel, Richard D. ( January 2023 , IEEE Transactions on Communications)

   This paper applies probabilistic amplitude shaping (PAS) to cyclic redundancy check (CRC)-aided tail-biting trellis-coded modulation (TCM). CRC-TCM-PAS produces practical codes for short block lengths on the additive white Gaussian noise (AWGN) channel. In the transmitter, equally likely message bits are encoded by a distribution matcher (DM) generating amplitude symbols with a desired distribution. A CRC is appended to the sequence of amplitude symbols, and this sequence is then encoded and modulated by TCM to produce real-valued channel input signals. This paper proves that the sign values produced by the TCM are asymptotically equally likely to be positive or negative. The CRC-TCM-PAS scheme can thus generate channel input symbols with a symmetric capacity-approaching probability mass function. The paper provides an analytical upper bound on the frame error rate of the CRC-TCM-PAS system over the AWGN channel. This FER upper bound is the objective function used for jointly optimizing the CRC and convolutional code. Additionally, this paper proposes a multi-composition DM, which is a collection of multiple constant-composition DMs. The optimized CRC-TCM-PAS systems achieve frame error rates below the random coding union (RCU) bound in AWGN and outperform the short-blocklength PAS systems with various other forward error correction codes studied in [2]. 

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

   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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3. Achieving Short-Blocklength RCU bound via CRC List Decoding of TCM with Probabilistic Shaping

   https://doi.org/10.1109/ICC45855.2022.9838498  

   Wang, L. ; Song, D. ; Areces, F. ; Wesel, R.D. ( May 2022 , 2022 International Conference on Communications)

   This paper applies probabilistic amplitude shaping (PAS) to a cyclic redundancy check (CRC) aided trellis coded modulation (TCM) to achieve the short-blocklength random coding union (RCU) bound. In the transmitter, the equally likely message bits are first encoded by distribution matcher to generate amplitude symbols with the desired distribution. The binary representations of the distribution matcher outputs are then encoded by a CRC. Finally, the CRC-encoded bits are encoded and modulated by Ungerboeck's TCM scheme, which consists of a k/(k+1) systematic tail-biting convolutional code and a mapping function that maps coded bits to channel signals with capacity-achieving distribution. This paper proves that, for the proposed transmitter, the CRC bits have uniform distribution and that the channel signals have symmetric distribution. In the receiver, the serial list Viterbi decoding (S-LVD) is used to estimate the information bits. Simulation results show that, for the proposed CRC-TCM-PAS system with 87 input bits and 65-67 8-AM coded output symbols, the decoding performance under additive white Gaussian noise channel achieves the RCU bound with properly designed CRC and convolutional codes. 

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4. 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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5. A perspective on massive random-access

   Polyanskiy, Yury ( June 2017 , 2017 IEEE Int. Symp. Inf. Theory (ISIT))

   This paper discusses the contemporary problem of providing multiple-access (MAC) to a massive number of uncoordinated users. First, we define a random-access code for Ka-user Gaussian MAC to be a collection of norm-constrained vectors such that the noisy sum of any Ka of them can be decoded with a given (suitably defined) probability of error. An achievability bound for such codes is proposed and compared against popular practical solutions: ALOHA, coded slotted ALOHA, CDMA, and treating interference as noise. It is found out that as the number of users increases existing solutions become vastly energy-inefficient. Second, we discuss the asymptotic (in blocklength) problem of coding for a K-user Gaussian MAC when K is proportional to blocklength and each user’s payload is fixed. It is discovered that the energy-per-bit vs. spectral efficiency exhibits a rather curious tradeoff in this case. 

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