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1. 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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2. Shaped TCM with List Decoding that Exceeds the RCU Bound by Optimizing a Union Bound on FER

   https://doi.org/10.1109/GLOBECOM48099.2022.10000695  

   Song, Dan; Areces, Felipe; Wang, Linfang; Wesel, Richard (December 2022, GLOBECOM 2022 - 2022 IEEE Global Communications Conference)

   This paper derives a union bound on the frame error rate (FER) of a probabilistic amplitude shaping (PAS) system which uses a CRC-aided, rate −k/k+1 , systematic, recursive trellis-coded modulation (TCM). A tail-biting convolutional code (TBCC) provides the feed-forward error correction (FEC) code for the TCM. The system is referred as CRC-TCM-PAS [1]. In order to derive the union bound, we first prove that the concatenation of a CRC and a rate −k/k+1 convolutional code is equivalent to a new convolutional code. Then, we give the generating function of the new convolutional code using Biglieri's product-state-diagram approach. A union bound can be calculated using the generating function. Simulation results show that the derived union bound is tight in the high signal-to-noise ratio (SNR) regime and can be used to design the convolutional and CRC codes. Simulation results also show that the optimized CRC-TCM-PAS system exceeds the random coding union (RCU) bound and outperforms the PAS systems with various FEC codes studied in [2] for the same number of input bits and the same transmission rate. 

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3. CRC-Aided Short Convolutional Codes and RCU Bounds for Orthogonal Signaling

   https://doi.org/10.1109/GLOBECOM48099.2022.10001459  

   King, Jacob; Ryan, William; Wesel, Richard D. (December 2022, GLOBECOM 2022 - 2022 IEEE Global Communications Conference)

   We extend earlier work on the design of convolutional code-specific CRC codes to Q -ary alphabets, with an eye toward Q -ary orthogonal signaling. Starting with distance-spectrum optimal, zero-terminated, Q -ary convolutional codes, we design Q -ary CRC codes so that the CRC/convolutional concatenation is distance-spectrum optimal. The Q -ary code symbols are mapped to a Q -ary orthogonal signal set and sent over an AWGN channel with noncoherent reception. We focus on Q=4 , rate-1/2 convolutional codes in our designs. The random coding union bound and normal approximation are used in earlier works as benchmarks for performance for distance-spectrum-optimal convolutional codes. We derive a saddlepoint approximation of the random coding union bound for the coded noncoherent signaling channel, as well as a normal approximation for this channel, and compare the performance of our codes to these limits. Our best design is within 0.6 dB of the RCU bound at a frame error rate of 10 −4 . 

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4. High-Rate Convolutional Codes with CRC-Aided List Decoding for Short Blocklengths

   https://doi.org/10.1109/ICC45855.2022.9839021  

   Sui, W.; Yang, H.; Towell, B.; Asmani, A.; Wesel, R. D. (May 2022, 2022 International Conference on Communications)

   Recently, rate-1/ω zero-terminated and tail-biting convolutional codes (ZTCCs and TBCCs) with cyclic-redundancy-check (CRC)-aided list decoding have been shown to closely approach the random-coding union (RCU) bound for short blocklengths. This paper designs CRCs for rate-(ω−1)/ω CCs with short blocklengths, considering both the ZT and TB cases. The CRC design seeks to optimize the frame error rate (FER) performance of the code resulting from the concatenation of the CRC and the CC. Utilization of the dual trellis proposed by Yamada et al. lowers the complexity of CRC-aided serial list Viterbi decoding (SLVD) of ZTCCs and TBCCs. CRC-aided SLVD of the TBCCs closely approaches the RCU bound at a blocklength of 128. 

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5. CRC-Aided List Decoding of Convolutional and Polar Codes for Short Messages in 5G

   https://doi.org/10.1109/ICC45855.2022.9838726  

   King, J.; Kwon, A.; Yang, H.; Ryan, W.; Wesel, R. D. (May 2022, 2022 International Conference on Communications)

   This paper explores list decoding of convolutional and polar codes for short messages such as those found in the 5G physical broadcast channel. A cyclic redundancy check (CRC) is used to select a codeword from a list of likely codewords. One example in the 5G standard encodes a 32-bit message with a 24-bit CRC and a 512-bit polar code with additional bits added by repetition to achieve a very low rate of 32/864. This paper shows that optimizing the CRC length improves the Eb/N0 performance of this polar code, where Eb/N0 is the ratio of the energy per data bit to the noise power spectral density. Furthermore, even better Eb/N0 performance is achieved by replacing the polar code with a tail-biting convolutional code (TBCC) with a distance-spectrum-optimal (DSO) CRC. This paper identifies the optimal CRC length to minimize the frame error rate (FER) of a rate-1/5 TBCC at a specific value of Eb/N0. We also show that this optimized TBCC/CRC can attain the same excellent Eb/N0 performance with the very low rate of 32/864 of the 5G polar code, where the low rate is achieved through repetition. We show that the proposed TBCC/CRC concatenated code outperforms the PBCH polar code described in the 5G standard both in terms of FER and decoding run time. We also explore the tradeoff between undetected error rate and erasure rate as the CRC size varies. 

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