Search for: All records

This paper presents 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 bounds for two cases: The first case considers VLSF codes with possibly infinite decoding times and zero error probability. The second case limits the maximum (finite) number of decoding times and specifies a maximum tolerable probability of error. Both new achievability bounds are proved by constructing a new VLSF code that employs systematic transmission of the first k message 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 et al. in 2016. We show that the backoff from capacity reduces to zero as the erasure probability decreases, thus giving a negative answer to the open question Devassy et al. posed on whether the 23.4% backoff to capacity at k=3 is fundamental to all BECs. For VLSF codes with finite decoding times, numerical evaluations show that the systematic transmission followed by random linear fountain coding performs better than random linear coding in terms of achievable rates. 

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. 

In this paper, we consider the problem of sequential transmission over the binary symmetric channel (BSC) with full, noiseless feedback. Naghshvar et al. proposed a one-phase encoding scheme, for which we refer to as the small-enough difference (SED) encoder, which can achieve capacity and Burnashev's optimal error exponent for symmetric binary-input channels. They also provided a non-asymptotic upper bound on the average blocklength, which implies an achievability bound on rates. However, their achievability bound is loose compared to the simulated performance of SED encoder, and even lies beneath Polyanskiy's achievability bound of a system limited to stop feedback. This paper significantly tightens the achievability bound by using a Markovian analysis that leverages both the submartingale and Markov properties of the transmitted message. Our new non-asymptotic lower bound on achievable rate lies above Polyanskiy's bound and is close to the actual performance of the SED encoder over the BSC. 

Building on the work of Horstein, Shayevitz and Feder, and Naghshvar et al., this paper presents algorithms for low-complexity sequential transmission of a k-bit message over the binary symmetric channel (BSC) with full, noiseless feedback. To lower complexity, this paper shows that the initial k binary transmissions can be sent before any feedback is required and groups messages with equal posteriors to reduce the number of posterior updates from exponential in k to linear in k. Simulation results demonstrate that achievable rates for this full, noiseless feedback system approach capacity rapidly as a function of average blocklength, faster than known finite-blocklength lower bounds on achievable rate with noiseless active feedback and significantly faster than finite-blocklength lower bounds for a stop feedback system. 

Cyclic redundancy check (CRC) codes combined with convolutional codes yield a powerful concatenated code that can be efficiently decoded using list decoding. To help design such systems, this paper presents an efficient algorithm for identifying the distance-spectrum-optimal (DSO) CRC polynomial for a given tail-biting convolutional code (TBCC) when the target undetected error rate (UER) is small. Lou et al. found that the DSO CRC design for a given zero-terminated convolutional code under low UER is equivalent to maximizing the undetected minimum distance (the minimum distance of the concatenated code). This paper applies the same principle to design the DSO CRC for a given TBCC under low target UER. Our algorithm is based on partitioning the tail-biting trellis into several disjoint sets of tail-biting paths that are closed under cyclic shifts. This paper shows that the tail-biting path in each set can be constructed by concatenating the irreducible error events (IEEs) and circularly shifting the resultant path. This motivates an efficient collection algorithm that aims at gathering IEEs, and a search algorithm that reconstructs the full list of error events with bounded distance of interest, which can be used to find the DSO CRC. Simulation results show that DSO CRCs can significantly outperform suboptimal CRCs in the low UER regime.