This paper presents a variable-length decision-feedback coding scheme that achieves high rates at short blocklengths. This scheme uses the reliability-output Viterbi algorithm (ROVA) to determine when the receiver's decoding estimate satisfies a given error constraint. We evaluate the performance of both terminated and tail-biting convolutional codes at average blocklengths less than 300 symbols, using the ROVA and the tail-biting ROVA, respectively. Comparing with recent results from finite-blocklength information theory, simulations for both the BSC and the AWGN channel show that the reliability-based decision-feedback scheme can surpass the random-coding lower bound on throughput for feedback codes at some blocklengths less than 100 symbols. This is true both when decoding after every symbol is permitted and when decoding is limited to a small number of increments. Finally, the performance of the reliability-based stopping rule with the ROVA is compared with retransmission decisions based on CRCs. For short blocklengths where the latency overhead of the CRC bits is severe, the ROVA-based approach delivers superior rates.

This paper studies variable-length coding with noise-less feedback for discrete memoryless channels. Yamamoto and Itoh's two-phase scheme achieves the optimal error-exponent, but due to the block-coding nature it is not optimal in the expansion of the message size logM. Polyanskiy et al. showed that with feedback, the back-off from capacity is logarithmic in the expected latency ℓ. The O(log ℓ) back-off is achieved by using an incremental redundancy (IR) scheme that only utilizes feedback to determine the stopping time. However, the achievable error-exponent of the IR scheme is not optimal. This paper shows that a two-phase coding scheme where each phase uses an IR scheme achieves the optimal error-exponent while maintaining an expansion on the message size that yields the O(log ℓ) back-off.

One advantage of feedback in a point-to-point memoryless channel is the reduction of the average blocklength required to approach capacity. This paper presents a communication system with feedback that uses carefully designed non-binary LDPC (NB-LDPC) codes and incremental transmissions to achieve 92–94% of the idealized throughput of rate-compatible sphere-packing with maximum-likelihood decoding (RCSP-ML) for average blocklengths of 150–450 bits. The system uses active feedback by carefully selecting each bit of additional incremental information to improve the reliability of the least reliable variable node. The system uses post processing in the decoder to further improve performance. The average blocklengths of 150–450 bits are small enough that feedback provides a throughput advantage but also large enough that overhead that might be associated with transmitter confirmation is more easily tolerated.

One advantage of incremental transmissions with feedback in point-to-point memoryless channels is a reduction in average blocklength required to approach capacity. This paper optimizes the size of each incremental transmission for non-binary (NB) LDPC codes to maximize throughput in VLFT and two-phase VLF settings. The optimization problem uses an approximation based on the inverse-Gaussian p.d.f. of the blocklength required for successful decoding. By using the optimized incremental transmission lengths (with an average blocklength of less than 500 bits), NB-LDPC codes for VLFT setting limited to 5 transmissions achieve a throughput greater than 96% of that obtained by an unlimited-transmission VLFT scheme with the same average blocklength. With a similar average blocklength, a two-phase VLF system limited to five transmissions (with optimized lengths) using the binary image of NB-LDPC codes achieves greater than 90% of the capacity of binary-input AWGN channel with SNR=2 dB. Two-phase VLF does not match the throughput of VLFT, but it is more practical than VLFT because it does not assume noiseless transmitter confirmation.

We present extensions to Raghavan and Baum's reliability-output Viterbi algorithm (ROVA) to accommodate tail-biting convolutional codes. These tail-biting reliability-output algorithms compute the exact word-error probability of the decoded codeword after first calculating the posterior probability of the decoded tail-biting codeword's starting state. One approach employs a state-estimation algorithm that selects the maximum a posteriori state based on the posterior distribution of the starting states. Another approach is an approximation to the exact tail-biting ROVA that estimates the word-error probability. A comparison of the computational complexity of each approach is discussed in detail. The presented reliability-output algorithms apply to both feedforward and feedback tail-biting convolutional encoders. These tail-biting reliability-output algorithms are suitable for use in reliability-based retransmission schemes with short blocklengths, in which terminated convolutional codes would introduce rate loss.

Theoretical analysis has long indicated that feedback improves the error exponent but not the capacity of single user memoryless channels. Recently Polyanskiy et al. studied the benefit of variable-length feedback with termination (VLFT) codes in the non-asymptotic regime. In that work, achievability is based on an infinite-length random code and decoding is attempted at every symbol. The coding rate backoff from capacity due to channel dispersion is greatly reduced with feedback, allowing capacity to be approached with surprisingly small expected latency. This paper is concerned with VLFT codes based on finite-length codes and decoding attempts only at certain specified decoding times. Note that with an underlying finite-length code, the transmitter may have to repeat code symbols. The penalties of using a finite block-length N and a sequence of specified decoding times are studied. This paper shows that properly scaling N with the expected latency can achieve the same performance up to second order terms as with N =∞. The penalty introduced by limiting the decoding is a constant term and hence the performance approaches capacity as expected latency increases as long as the interval between periodic decoding times grows sub-linearly with the expected latency.

This paper presents a reliability-based decoding scheme for variable-length coding with feedback and demonstrates via simulation that it can achieve higher rates than Polyanskiy et al.'s random coding lower bound for variable-length feedback (VLF) coding on both the BSC and AWGN channel. The proposed scheme uses the reliability output Viterbi algorithm (ROVA) to compute the word error probability after each decoding attempt, which is compared against a target error threshold and used as a stopping criterion to terminate transmission. The only feedback required is a single bit for each decoding attempt, informing the transmitter whether the ROVA-computed word-error probability is sufficiently low. Furthermore, the ROVA determines whether transmission/decoding may be terminated without the need for a rate-reducing CRC.

Recent work by Polyanskiy et al. and Chen et al. has excited new interest in using feedback to approach capacity with low latency. Polyanskiy showed that feedback identifying the first symbol at which decoding is successful allows capacity to be approached with surprisingly low latency. This paper uses Chen's rate-compatible sphere-packing (RCSP) analysis to study what happens when symbols must be transmitted in packets, as with a traditional hybrid ARQ system, and limited to relatively few (six or fewer) incremental transmissions. Numerical optimizations find the series of progressively growing cumulative block lengths that enable RCSP to approach capacity with the minimum possible latency. RCSP analysis shows that five incremental transmissions are sufficient to achieve 92% of capacity with an average block length of fewer than 101 symbols on the AWGN channel with SNR of 2.0 dB. The RCSP analysis provides a decoding error trajectory that specifies the decoding error rate for each cumulative block length. Though RCSP is an idealization, an example tail-biting convolutional code matches the RCSP decoding error trajectory and achieves 91% of capacity with an average block length of 102 symbols on the AWGN channel with SNR of 2.0 dB. We also show how RCSP analysis can be used in cases where packets have deadlines associated with them (leading to an outage probability).