Guessing Random Additive Noise Decoding of Network Coded Data Transmitted over Burst Error Channels

TL;DR

This paper introduces transversal GRAND (T-GRAND), a burst-error aware extension of GRAND that operates alongside packet-level RLC decoding to repair erroneous coded packets in memoryful channels. By ordering error vectors according to their likelihood under a simplified Gilbert-Elliott model, T-GRAND outperforms syndrome decoding in terms of decoding probability and average transmissions, especially as bit error probability and burst length grow. The approach combines a reduced parity-check framework with a trace-based sorting procedure to manage complexity, and demonstrates substantial gains in decoding success and completion time in bursty channels relevant to network coding. The results support practical benefits for AL-FEC in wireless and vehicular networks, where burst errors and limited interleaving are common, and point to avenues for extensions to higher-order finite fields and hardware implementations.

Abstract

We consider a transmitter that encodes data packets using network coding and broadcasts coded packets. A receiver employing network decoding recovers the data packets if a sufficient number of error-free coded packets are gathered. The receiver does not abandon its efforts to recover the data packets if network decoding is unsuccessful; instead, it employs syndrome decoding (SD) in an effort to repair erroneous received coded packets, and then reattempts network decoding. Most decoding techniques, including SD, assume that errors are independently and identically distributed within received coded packets. Motivated by the guessing random additive noise decoding (GRAND) framework, we propose transversal GRAND (T-GRAND): an algorithm that exploits statistical dependence in the occurrence of errors, complements network decoding and recovers all data packets with a higher probability than SD. T-GRAND examines error vectors in order of their likelihood of occurring and altering the transmitted packets. Calculation and sorting of the likelihood values of all error vectors is a simple but computationally expensive process. To reduce the complexity of T-GRAND, we take advantage of the properties of the likelihood function and develop an efficient method, which identifies the most likely error vectors without computing and ordering all likelihood values.

Figures (10)

• Figure 1: The simplified Gilbert-Elliott channel model, where $0$ and $1$ represent the 'good' state and the 'bad' state, respectively.
• Figure 2: Example of the derivation of column $b$ of $\hat{\mathbf{E}}_{\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{R}\mkern-1.5mu}\mkern 1.5mu}$ using Algorithm \ref{['alg:TGRAND1']} when the previous column of $\hat{\mathbf{E}}_{\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{R}\mkern-1.5mu}\mkern 1.5mu}$ contains $L_0=2$ zeros and $L_1=3$ ones. The transition probabilities are set to $p_{01}=0.1$ and $p_{10}=0.4$.
• Figure 3: Pictorial representation of how TraceSortedProb pinpoints the entries of matrix $\mathbf{\Phi}$, from the smallest to the largest, for $p_{01}<\tfrac{1}{2},$$p_{10}<\tfrac{1}{2}$ and $p_{01} > p_{10}$. Equivalently, $\alpha_0>0$, $\alpha_1>0$ and $\alpha_0 < \alpha_1$. (a) A fragment of the matrix $\mathbf{\Phi}$ with the plane above visualizing the value of its elements. The smallest entry of $\mathbf{\Phi}$, which corresponds to the largest probability value, is at $(\ell_0^{(1)},\ell_1^{(1)})=(0,0)$ and is depicted by a large blue ball. Two candidates for the second smallest value are shown as small red balls at $(\ell_0,\ell_1)=(1,0)$ and $(\ell_0,\ell_1)=(0,1)$, where $\varphi(\ell_0, \ell_1)$ takes the values $\alpha_0$ and $\alpha_1$, respectively. Since $\alpha_0 < \alpha_1$, the second smallest entry of $\mathbf{\Phi}$ is at $(\ell_0^{(2)},\ell_1^{(2)})=(1,0)$. (b) A larger fragment of matrix $\mathbf{\Phi}$. The shaded blue region represents the set $\mathcal{L}$ of the $\ell$ smallest entries of $\mathbf{\Phi}$. The largest element in $\mathcal{L}$ is depicted by a large blue ball. The candidates for the $(\ell+1)$-th smallest entry are the corners of $\bar{\mathcal{L}}$, shown as small red balls. Labels on candidate elements represent the values $\varphi(\ell_0,\ell_1)=\alpha_0\ell_0+\alpha_1\ell_1$.
• Figure 4: Example demonstrating the operation of TraceSortedProb for $p_{01}=0.4$, $p_{10}=0.3$, $L_0=3$ and $L_1=3$. Based on these values, $\alpha_0=0.59$ and $\alpha_1=1.22$ (displayed values have been rounded up to two decimal places). The entries of $\mathbf{\Phi}$ as well as a path that begins at the smallest entry and ends at the fifth smallest entry of $\mathbf{\Phi}$ are presented in (a) for reference. The process of TraceSortedProb for identifying the five smallest entries of $\mathbf{\Phi}$, without first calculating all entries and then ordering them, is described in (b)-(f). Blue balls of any size identify corners of $\mathcal{L}$, while red balls of any size represent corners of $\bar{\mathcal{L}}$. A large blue ball marks the coordinate pair that has been added last to $\mathcal{L}$, as it points to the $\ell$-th smallest value of $\mathbf{\Phi}$, for $\ell=1$ in (b) to $\ell=5$ in (f). Labels on red balls represent the values of $\varphi(\ell_0,\ell_1)$ at these points. A large red ball pinpoints the corner of $\bar{\mathcal{L}}$ that is chosen for a value of $\ell$ and changes to a large blue ball after it is added to $\mathcal{L}$ for the next value of $\ell$. Observe that the large blue balls in (b)-(f) occupy the same coordinates and in the same order as the coordinates of the five smallest entries in (a).
• Figure 5: Comparison of the decoding probabilities of RLC decoding, RLC decoding with SD, and RLC decoding with T-GRAND for $K=10$ source packets, $N$ transmitted coded packets, where $N=10,\ldots,20$, bit error probability $\varepsilon\in\{0.01,0.03,0.05\}$, average length of error bursts $\Lambda=4$ and packet length $B=64$.