This paper uses convolutional codes (CCs) with distance-spectrum optimal (DSO) cyclic redundancy checks (CRCs) and the serial list Viterbi algorithm (S-LVA) to approach the random coding union (RCU) bound with low decoding complexity at the target FER. We show, for example, that a 64-state zero-terminated CC with a DSO CRC can achieve performance within 0.5 dB of the RCU bound for information blocklength k=64 at FER of 10^{-3}. We also show that a tail-biting CC with a DSO CRC can achieve even better performance, within 0.05 dB of the RCU bound at FER of 10^{-4} for a 256-state CC with k=64. This paper provides analysis of decoding complexity, which for S-LVA depends on the expected list size. We show that if the target FER is low enough, the expected list size approaches one so that the average complexity of S-LVA approaches that of standard soft Viterbi on the CC, i.e., with no list decoding. We also provide DSO CRCs for CCs with k=64 and rates of 1/2, 1/3, 1/6 and 1/12 for the 5G new radio control channel and compare their performance with polar codes.

List-Decoded Tail-Biting Convolutional Codes with Distance-Spectrum Optimal CRCS for 5G

This paper identifies convolutional codes (CCs) used in conjunction with a CC-specific cyclic redundancy check (CRC) code as a promising paradigm for short blocklength codes. The resulting CRC-CC concatenated code naturally permits the use of the serial list Viterbi decoding (SLVD) to achieve maximum-likelihood decoding. The CC of interest is of rate-$1/\omega$ and is either zero-terminated (ZT) or tail-biting (TB). For CRC-CC concatenated code designs, we show how to find the optimal CRC polynomial for a given ZTCC or TBCC. Our complexity analysis reveals that SLVD decoding complexity is a function of the terminating list rank, which converges to one at high SNR. This behavior allows the performance gains of SLVD to be achieved with a small increase in average complexity at the SNR operating point of interest. With a sufficiently large CC constraint length, the performance of CRC-CC concatenated code under SLVD approaches the random-coding union (RCU) bound as the CRC size is increased while average decoding complexity does not increase significantly. TB encoding further reduces the backoff from the RCU bound by avoiding the termination overhead. As a result, several CRC-TBCC codes outperform the RCU bound at moderate SNR values while permitting decoding with relatively low complexity.

CRC-Aided List Decoding of Convolutional Codes in the Short Blocklength Regime.

We consider the concatenation of a convolutional code (CC) with an optimized cyclic redundancy check (CRC) code as a promising paradigm for good short blocklength codes. The resulting CRC-aided convolutional code naturally permits the use of serial list Viterbi decoding (SLVD) to achieve maximum-likelihood decoding. The convolutional encoder of interest is of rate-$1/\omega$ and the convolutional code is either zero-terminated (ZT) or tail-biting (TB). The resulting CRC-aided convolutional code is called a CRC-ZTCC or a CRC-TBCC. To design a good CRC-aided convolutional code, we propose the distance-spectrum optimal (DSO) CRC polynomial. A DSO CRC search algorithm for the TBCC is provided. Our analysis reveals that the complexity of SLVD is governed by the expected list rank which converges to $1$ at high SNR. This allows a good performance to be achieved with a small increase in complexity. In this paper, we focus on transmitting $64$ information bits with a rate-$1/2$ convolutional encoder. For a target error probability $10^{-4}$, simulations show that the best CRC-ZTCC approaches the random-coding union (RCU) bound within $0.4$ dB. Several CRC-TBCCs outperform the RCU bound at moderate SNR values. 

/pdf/crc-aided-list-decoding-of-convolutional-codes-in-the-short-1v3fls6k.pdf

CRC-Aided List Decoding of Convolutional Codes in the Short Blocklength Regime

This paper studies the joint design of optimal convolutional codes (CCs) and CRC codes when serial list Viterbi algorithm (S-LVA) is employed in order to achieve the target frame error rate (FER). We first analyze the S-LVA performance with respect to SNR and list size, repsectively, and prove the convergence of the expected number of decoding attempts when SNR goes to the extreme. We then propose the coded channel capacity as the criterion to jointly design optimal CC-CRC pair and optimal list size and show that the optimal list size of S-LVA is always the cardinality of all possible CCs. With the maximum list size, we choose the design metric of optimal CC-CRC pair as the SNR gap to random coding union (RCU) bound and the optimal CC-CRC pair is the one that achieves a target SNR gap with the least complexity. Finally, we show that a weaker CC with a strong optimal CRC code could be as powerful as a strong CC with no CRC code.

Joint Design of Convolutional Code and CRC under Serial List Viterbi Decoding.

This paper presents the implementation result of List-of-2 soft-decision Viterbi decoder on FPGA platform. It was to be used as the inner decoder of a special designed concatenated code, of which the outer decoder is Vector Symbol Decoder (VSD). Since list-of-2 Viterbi was not the final decoding stage, it was not required to provide low decoding failure probability. Therefore, only a small list of 2 was enough. This use of list inner decoder is useful to VSD only if the second inner decoded sequence is correct when the first one was wrong. The result shows that this probability of interest is high for low-SNR AWGN channel and BFSK scheme. Therefore, List-of-2 Viterbi would be attractive for VSD for this channel. The operation of this lab prototype was tested by comparing its decoding failure probabilities with the results from C++. The results showed that they were almost the same, which proved that the lab prototype worked properly.

http://ieeexplore.ieee.org/iel5/5483296/5491394/05491620.pdf

Lab Prototype of a List-of-2 Viterbi decoder: A diversity inner decoder for the outer Vector Symbol Decoder

In this study, we propose a list single-wrong turn (SWT) convolutional decoding algorithm that is computationally less complex than the serial list Viterbi algorithm (SLVA). We present the simulation results for the performance of the proposed algorithm for Gaussian and Rayleigh fading channels using one of the channel encoding schemes adopted in GSM narrowband adaptive multi-rate (AMR) speech application. It has been observed that the performance of the proposed algorithm is very close to that of the list Viterbi algorithm, when the number of searches is limited to best 3 sequences. Moreover, the complexity of the proposed algorithm is considerably less than list VA. Therefore, we believe that the proposed scheme is an attractive alternative to the list VA algorithm.

A reduced complexity list single-wrong-turn (SWT) Viterbi decoding algorithm

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A reduced complexity list single-wrong-turn (SWT) Viterbi decoding algorithm