Sequential Decoding of Multiple Traces Over the Syndrome Trellis for Synchronization Errors

TL;DR

This work tackles synchronization errors in channels by proposing a joint sequential decoding approach on the syndrome trellis for multiple sequences. It adapts the stack algorithm to operate over the syndrome tree and extends it bidirectionally to reduce erasures, enabling efficient decoding of high-rate convolutional codes in short blocklength regimes. For certain noise and code-rate regimes, jointly decoding multiple sequences on the syndrome trellis with the stack algorithm can outperform separate-BCJR decoding, while offering substantial reductions in decoding complexity, especially at moderate noise levels and with a modest number of sequences. The approach holds practical significance for applications like DNA data storage, where high-rate codes and joint decoding of multiple noisy observations are common.

Abstract

Standard decoding approaches for convolutional codes, such as the Viterbi and BCJR algorithms, entail significant complexity when correcting synchronization errors. The situation worsens when multiple received sequences should be jointly decoded, as in DNA storage. Previous work has attempted to address this via separate-BCJR decoding, i.e., combining the results of decoding each received sequence separately. Another attempt to reduce complexity adapted sequential decoders for use over channels with insertion and deletion errors. However, these decoding alternatives remain prohibitively expensive for high-rate convolutional codes. To address this, we adapt sequential decoders to decode multiple received sequences jointly over the syndrome trellis. For the short blocklength regime, this decoding strategy can outperform separate-BCJR decoding under certain channel conditions, in addition to reducing decoding complexity. To mitigate the occurrence of a decoding timeout, formally called erasure, we also extend this approach to work bidirectionally, i.e., deploying two independent stack decoders that simultaneously operate in the forward and backward directions.

Figures (6)

• Figure 1: Allowed transitions in the state machine model for the insertion, deletion, and substitution channel daveyReliableCommunicationChannels2001.
• Figure 2: Syndrome trellis of the $[3,2]$ convolutional code of length $N=9$ in Example \ref{['eg::cc']}. Dashed and solid edges emerging from a node at level $l$ represent codeword bits $x_{l+1}=0$ and $x_{l+1}=1$, respectively. For longer codes, the trellis consists of successive repetitions of the block within the dashed box. The final three stages show the termination phase. Levels $0, 1, 3, 4$ are information levels while levels $2, 5, 6,7, 8$ are parity levels. The green, blue, and red stages indicate information bits, parity bits, and termination bits, respectively.
• Figure 3: Part of the joint channel and code tree of the $[3,2]$ convolutional code from Example \ref{['eg::cc']}. Dashed and solid lines indicate outputs $0$ and $1$, respectively. Each node represents a pair of a syndrome state and a drift state $({\boldsymbol s}_l, d_j)$. The notations $S_0, \ldots, S_6$ represent specific realizations of a syndrome state. For ease of illustration, we restrict the net drift per branch to $\{-1, 0, 1\}$. The green and blue stages indicate information bits and parity bits, respectively.
• Figure 4: BER versus $P_{\mathrm{d}}$ when $P_{\mathrm{i}}=P_{\mathrm{s}}=0$ and $M=2$. CC1 and CC2 have lengths $N=126$ and $N=139$, respectively.
• Figure 5: BER versus $P_{\mathrm{i}} = P_{\mathrm{d}}$ when $P_{\mathrm{s}}=0$ for CC2 with varying number of received sequences, $M$. The codewords are of length $N=139$.

Theorems & Definitions (2)

• Example 1
• Example 2