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Posterior matching uses variable-length encoding of the message controlled by noiseless feedback of the received symbols to achieve high rates for short average blocklengths. Traditionally, the feedback of a received symbol occurs before the next symbol is transmitted. The transmitter optimizes the next symbol transmission with full knowledge of every past received symbol. To move posterior matching closer to practical communication, this paper seeks to constrain how often feedback can be sent back to the transmitter. We focus on reducing the frequency of the feedback while still maintaining the high rates that posterior matching achieves with feedback after every symbol. As it turns out, the frequency of the feedback can be reduced significantly with no noticeable reduction in rate.

In this paper, we present new achievability bounds on the maximal achievable rate of variable-length stop-feedback (VLSF) codes operating over a binary erasure channel (BEC) at a fixed message size $M = 2^k$. We provide new bounds for VLSF codes with zero error, infinite decoding times and with nonzero error, finite decoding times. Both new achievability bounds are proved by constructing a new VLSF code that employs systematic transmission of the first $k$ bits followed by random linear fountain parity bits decoded with a rank decoder. For VLSF codes with infinite decoding times, our new bound outperforms the state-of-the-art result for BEC by Devassy \emph{et al.} in 2016. We also give a negative answer to the open question Devassy \emph{et al.} put forward on whether the $23.4\%$ backoff to capacity at $k = 3$ is fundamental. For VLSF codes with finite decoding times, numerical evaluations show that the achievable rate for VLSF codes with a moderate number of decoding times closely approaches that for VLSF codes with infinite decoding times., Comment: 7 pages, double column, 4 figures; comments are welcome! changes in v2: corrected 2 typos in v1. arXiv admin note: substantial text overlap with arXiv:2205.15399

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., Comment: submitted to ISTC 2023

Motivated by the need to communicate short control messages in 5G and beyond, this paper carefully designs codes for cyclic redundancy check (CRC)-aided list decoding of tail-biting convolutional codes (TBCCs) and polar codes. Both codes send a 32-bit message using an 11-bit CRC and 512 transmitted bits. We aim to provide a careful, fair comparison of the error performance and decoding complexity of polar and TBCC techniques for a specific case. Specifically, a TBCC is designed to match the rate of a (512, 43) polar code, and optimal 11-bit CRCs for both codes are designed. The paper examines the distance spectra of the polar and TBCC codes, illuminating the different distance structures for the two code types. We consider both adaptive and non-adaptive CRC-aided list decoding schemes. For polar codes, an adaptive decoder must start with a larger list size to avoid an error floor. For rate-32/512 codes with an 11-bit CRC, the optimized CRC-TBCC design achieves a lower total failure rate than the optimized CRC-polar design. Simulations showed that the optimized CRC-TBCC design achieved significantly higher throughput than the optimized CRC-polar design, so that the TBCC solution achieved a lower total failure rate while requiring less computational complexity., Comment: First revision submitted to IEEE Transactions on Communications

Recently, rate-1/n zero-terminated (ZT) and tail-biting (TB) convolutional codes (CCs) 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 CRC polynomials for rate- (n-1)/n ZT and TB CCs with short blocklengths. This paper considers both standard rate-(n-1)/n CC polynomials and rate- (n-1)/n designs resulting from puncturing a rate-1/2 code. The CRC polynomials are chosen to maximize the minimum distance d_min and minimize the number of nearest neighbors A_(d_min) . For the standard rate-(n-1)/n codes, utilization of the dual trellis proposed by Yamada et al. lowers the complexity of CRC-aided serial list Viterbi decoding (SLVD). CRC-aided SLVD of the TBCCs closely approaches the RCU bound at a blocklength of 128. This paper compares the FER performance (gap to the RCU bound) and complexity of the CRC-aided standard and punctured ZTCCs and TBCCs. This paper also explores the complexity-performance trade-off for three TBCC decoders: a single-trellis approach, a multi-trellis approach, and a modified single-trellis approach with pre-processing using the wrap around Viterbi algorithm., Comment: arXiv admin note: substantial text overlap with arXiv:2111.07929

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 $10^{-4}$., Comment: GLOBECOM 2022 camera-ready version

Incremental redundancy with ACK/NACK feedback produces a variable-length stop-feedback (VLSF) code constrained to have $m$ decoding times, with an ACK/NACK feedback to the transmitter at each decoding time. This paper focuses on the numerical evaluation of the maximal achievable rate of random VLSF codes as a function of $m$ for the binary-input additive white Gaussian noise channel, binary symmetric channel, and binary erasure channel (BEC). Leveraging Edgeworth and Petrov expansions, we develop tight approximations to the tail probability of length-$n$ cumulative information density that are accurate for any blocklength $n$. We reduce Yavas et al.'s non-asymptotic achievability bound on VLSF codes with $m$ decoding times to an integer program of minimizing the upper bound on the average blocklength subject to the average error probability, minimum gap, and integer constraints. We develop two distinct methods to solve this program. Numerical evaluations show that Polyanskiy's achievability bound for VLSF codes, which assumes $m = \infty$, can be approached with a small $m$ for all three channels. For BEC, we consider systematic transmission followed by random linear fountain coding. This allows us to obtain a new achievability bound stronger than a previous bound and new VLSF codes whose rate further outperforms Polyanskiy's bound., Comment: 19 pages, 12 figures; submitted to IEEE Transactions on Communications. An earlier version of this work was published at ISIT 2022. Comments are welcome!

In this paper, we are interested in the performance of a variable-length stop-feedback (VLSF) code with $m$ optimal decoding times for the binary-input additive white Gaussian noise channel. We first develop tight approximations on the tail probability of length-$n$ cumulative information density. Building on the work of Yavas \emph{et al.}, for a given information density threshold, we formulate the integer program of minimizing the upper bound on average blocklength over all decoding times subject to the average error probability, minimum gap and integer constraints. Eventually, minimization of locally minimum upper bounds over all thresholds will yield the globally minimum upper bound and this is called the two-step minimization. For the integer program, we present a greedy algorithm that yields possibly suboptimal integer decoding times. By allowing a positive real-valued decoding time, we develop the gap-constrained sequential differential optimization (SDO) procedure that sequentially produces the optimal, real-valued decoding times. We identify the error regime in which Polyanskiy's scheme of stopping at zero does not improve the achievability bound. In this error regime, the two-step minimization with the gap-constrained SDO shows that a finite $m$ suffices to attain Polyanskiy's bound for VLSF codes with $m = \infty$., Comment: 8 pages, 4 figures; fixed some errors in v1 and added some new results; a short version of this preprint was submitted to ISIT 2022

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 $E_b/N_0$ performance of this polar code, where $E_b/N_0$ is the ratio of the energy per data bit to the noise power spectral density. Furthermore, even better $E_b/N_0$ 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 $E_b/N_0$. We also show that this optimized TBCC/CRC can attain the same excellent $E_b/N_0$ 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., Comment: 6 pages, 8 figures; this preprint is accepted for publication at the 2022 IEEE International Conference on Communications (ICC); camera-ready version to be updated

In this paper, we consider the problem of variable-length coding over the class of memoryless binary asymmetric channels (BACs) with noiseless feedback, including the binary symmetric channel (BSC) as a special case. In 2012, Naghshvar et al. introduced an encoding scheme, which we refer to as the small-enough-difference (SED) encoder, which asymptotically achieves both capacity and Burnashev's optimal error exponent for symmetric binary-input channels. Building on the work of Naghshvar et al., this paper extends the SED encoding scheme to the class of BACs and develops a non-asymptotic upper bound on the average blocklength that is shown to achieve both capacity and the optimal error exponent. For the specific case of the BSC, we develop an additional non-asymptotic bound using a two-phase analysis that leverages both a submartingale synthesis and a Markov chain time of first passage analysis. For the BSC with capacity $1/2$, both new achievability bounds exceed the achievability bound of Polyanskiy et al. for a system limited to stop-feedback codes., Comment: 18 pages; submitted to IEEE Transactions on Information Theory

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 $\frac{k_0}{k_0+1}$ 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.

Recently, rate-$1/\omega$ 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-$(\omega-1)/\omega$ 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 \emph{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 $128$., Comment: 6 pages; submitted to 2022 IEEE International Conference on Communications (ICC 2022)