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Maximum-likelihood (ML) decoding of tail-biting convolutional codes (TBCCs) with S = 2^v states traditionally requires a separate S-state trellis for each of the S possible starting/ending states, resulting in complexity proportional to S^2. Lower-complexity ML decoders for TBCCs have complexity proportional to S log S. This high complexity motivates the use of the wrap-around Viterbi algorithm, which sacrifices ML performance for complexity proportional to S. This paper presents an ML decoder for TBCCs that uses list decoding to achieve an average complexity proportional to S at operational signal-to-noise ratios where the expected list size is close to one. The new decoder uses parallel list Viterbi decoding with a progressively growing list size operating on a single S-state trellis. Decoding does not terminate until the most likely tailbiting codeword has been identified. This approach is extended to ML decoding of tail-biting convolutional codes concatenated with a cyclic redundancy check code as explored recently by Yang et al. and King et al. Constraining the maximum list size further reduces complexity but sacrifices guaranteed ML performance, increasing errors and introducing erasures. 

An algorithm is proposed to encode low-density parity-check (LDPC) codes into codewords with a non-uniform distribution. This enables power-efficient signalling for asymmetric channels. We show gains of 0.9 dB for additive white Gaussian noise (AWGN) channels with on-off keying modulation using 5G LDPC codes. 

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. 

List Viterbi decoders are a very effective way to improve the performance of block codes in combination with an error detection outer code. In this work, we combine an efficient serial list Viterbi decoder design with an existing serially concatenated, convolutionally-encoded, pulse position modulated code (SCPPM) used in space communication, that exhibits poor performance because of an error floor. The SCPPM code features a 32-bit CRC that provides powerful error detection capability and an outer four-state convolutional code that makes it suitable for a list Viterbi decoder. The system’s code is very long, consisting of 15, 120 bits, which renders a high complexity decoder impractical, while the high error detection allows for a list decoder with very low undetected error probability. We use a very efficient list Viterbi decoder algorithm to avoid most of the redundant operations to produce low complexity serial list Viterbi decoder. The combined system reduces the error floor, moderately for the original version of the system, and completely suppresses it when the code length is increased to four times longer. 

This paper introduces a mutual information (MI) maximization paradigm that adapts the locations and probabilities of write levels to iteratively increase the mutual information of the weakest bit channel and hence improve the reliability of its corresponding page. In this way, we seek a constellation of write levels that delivers the same amount of mutual information to the bit channel for each page, so that all pages are equally reliable. For simplicity, we consider the example of TLC Flash with an additive white Gaussian noise (AWGN) channel model, but the principle may be applied to denser cells and more realistic channel models. 

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. 

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. 

This paper compares the accuracy and complexity of Raghavan and Baum’s Reliability Output Viterbi Algorithm (ROVA), Polyanskiy’s accumulated information density (AID), and Fricke and Hoeher’s lower complexity approximation of ROVA. It turns out that AID is far less accurate than ROVA in practice. This paper proposes codeword information density (CID), which modifies AID to improve its accuracy and leads to a lower-complexity implementation of ROVA. The paper includes an analytical expression for the random variable describing the correct decoding probability computed by ROVA and uses this expression to characterize how the probabilities of correct decoding, undetected error, and negative acknowledgement behave as a function of the selected threshold for reliable decoding. This paper examines both the complexity and the simulation time of ROVA, CID, AID, and the Fricke and Hoeher approximation to ROVA. This paper also derives an expression for the union bound on the frame error rate for zero-terminated trellis codes with punctured symbols and uses it to optimize the order of symbol transmission in an incremental retransmission scheme. This paper concludes by comparing the performance of an incremental retransmission scheme using ROVA as a stopping condition to one that uses a CRC as a stopping condition.