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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]. 

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 . 

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 dmin . 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. 

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 to the tail probability of length-n cumulative information density. Building on the work of Yavas 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 optimal upper bounds over all thresholds yields the globally minimum upper bound and the above method is called the two-step minimization. Relaxing to allow positive real-valued decoding times activates the gap constraint. We develop gap-constrained sequential differential optimization (SDO) procedure to find the optimal, gap-constrained, real-valued decoding times. In the error regime of practical interest, Polyanskiy's scheme of stopping at zero does not help. In this region, the achievability bounds estimated by the two-step minimization and gap-constrained SDO show that Polyanskiy’s achievability bound for VLSF codes can be approached with a small number of decoding times. 

Traditional communication systems transmit a codeword only after all message bits are available at the transmitter. This paper joins Guo & Kostina and Lalitha et al. in developing approaches for causal encoding, where the transmitter may begin transmitting codeword symbols as soon as the first message bit arrives. Building on the posterior matching encoders of Horstein, Shayevitz & Feder, and Naghshvar et al., this paper extends our computationally efficient systematic encoder to progressively encode using only the message bits that are causally available. Systematic codes work well with posterior matching on a channel with feedback, and they provide an immediate benefit when causal encoding is employed instead of traditional encoding. Our algorithm captures additional gains in the interesting region where the transmission rate μ is higher than the source rate λ at which message bits become available. In this region, we improve performance further through the transmission of additional, non- systematic symbols before a traditional encoder would have even begun transmission. 

A status updating system is considered in which a variable length code is used to transmit messages to a receiver over a noisy channel. The goal is to optimize the codewords lengths such that successfully-decoded messages are timely. That is, such that the age-of-information (AoI) at the receiver is minimized. A hybrid ARQ (HARQ) scheme is employed, in which variable-length incremental redundancy (IR) bits are added to the originally-transmitted codeword until decoding is successful. With each decoding attempt, a non-zero processing delay is incurred. The optimal codewords lengths are analytically derived utilizing a sequential differential optimization (SDO) framework. The framework is general in that it only requires knowledge of an analytical expression of the positive feedback (ACK) probability as a function of the codeword length. 

Previously, dynamic-assignment Blahut-Arimoto (DAB) was used to find capacity-achieving probability mass functions (PMFs) for binomial channels and molecular channels. As it turns out, DAB can efficiently identify capacity-achieving PMFs for a wide variety of channels. This paper applies DAB to power-constrained (PC) additive white Gaussian Noise (AWGN) Channels and amplitude-constrained (AC) AWGN Channels.This paper modifies DAB to include a power constraint and finds low-cardinality PMFs that approach capacity on PC-AWGN Channels. While a continuous Gaussian PDF is well-known to be capacity-achieving on the PC-AWGN channel, DAB identifies low-cardinality PMFs within 0.01 bits of the mutual information provided by a Gaussian PDF. Recall the results of Ozarow and Wyner requiring a constellation cardinality of ⌈2 ^ (C+1) ⌉ to approach capacity C to within the asymptotic shaping loss of 1.53 dB at high SNR. PMF's found by DAB approach capacity with essentially no shaping loss with cardinality less than 2 ^ (C+1.2) . As expected, DAB's numerical approach identifies PMFs with better mutual information vs. SNR performance than the analytical approaches to finite-support constellations examined by Wu and Verdu. This paper also uses DAB to find capacity-achieving PMFs with small cardinality support sets for AC-AWGN Channels. The resulting evolution of capacity-achieving PMFs as a function of SNR is consistent with the approximate cardinality transition points of Sharma and Shamai.