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Efficient Error Control Decoder Architectures for Noncoherent Random Linear Network Coding

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Abstract

Random linear network coding is an efficient technique for disseminating information in networks, but it is highly susceptible to errors. Kötter-Kschischang (KK) codes and Mahdavifar-Vardy (MV) codes are two important families of subspace codes that provide error control in noncoherent random linear network coding. List decoding has been used to decode MV codes beyond half distance. Existing hardware implementations of the rank metric decoder for KK codes suffer from limited throughput, long latency and high area complexity. The interpolation-based list decoding algorithm for MV codes still has high computational complexity, and its feasibility for hardware implementations has not been investigated. In this paper we propose efficient decoder architectures for both KK and MV codes and present their hardware implementations. Two serial architectures are proposed for KK and MV codes, respectively. An unfolded decoder architecture, which offers high throughput, is also proposed for KK codes. The synthesis results show that the proposed architectures for KK codes are much more efficient than rank metric decoder architectures, and demonstrate that the proposed decoder architecture for MV codes is affordable.

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Correspondence to Jun Lin.

Appendices

Appendix A: Proof of the Lemma 3

Proof

Since f i (u i ) = 0 has two different roots, then the equation concerning u i is \(d^{(i)}_{0,i}+d^{(i)}_{1,i}u_i+d^{(i)}_{2,0}u_{i}^{2} = 0\) according to Eq. (13). Meanwhile, \(d^{(i)}_{0,j}= 0\) and \(d^{(i)}_{1,j}= 0\) for j < i. \(d^{(i)}_{1,i}\neq 0\), according to Lemma 2. Based on Eq. (10) and Eq. (11), we have

$$\begin{array}{@{}rcl@{}} \begin{array}{ll} dl^{(i+1)}(x,Y) = &c_{i+1,0}x^{[i]}+ \cdots +{d^{(i)}_{1,i}}^{[-1]}Y^{[i]}+ \cdots \\ &+ {d^{(i)}_{2,0}}^{[-1]}\left(Y\otimes Y^{[i+1]}\right)^{[0]}+\cdots, \end{array} \end{array}$$
(15)

where

$$c_{i+1,0} =\left(d^{(i)}_{0,i+1}+d^{(i+1)}_{1,i}u_{i,0}+d^{(i)}_{2,1}u_{i,0}^{2}\right)2^{[-1]}, {d^{(i)}_{1,i}}^{[-1]}\neq 0.$$

Let Y = u i + 1, 0 x for Eq. (15). We have

$$\begin{array}{@{}rcl@{}} \begin{array}{ll} dl^{(i+1)}(x,Y) = &c_{i+1,0}x^{[i]}+ \cdots +{d^{(i)}_{1,i}}^{[-1]}u_{i+1,0}x^{[i]}\\ &+ \cdots + {d^{(i)}_{2,0}}^{[-1]}u_{i+1,0}^2x^{[i+1]}+\cdots. \end{array} \end{array}$$
(16)

The equation about u i + 1, 0 is

$$f_{i+1,0}(u_{i+1,0})=c_{i+1,0}+{d^{(i)}_{1,i}}^{[-1]}u_{i+1,0}=0.$$
(17)

Apply the same arguments and we have

$$f_{i+1,1}(u_{i+1,1})=c_{i+1,1}+{d^{(i)}_{1,i}}^{[-1]}u_{i+1,1}=0.$$
(18)

Both f i + 1, 0(u i + 1, 0) = 0 and f i + 1, 1(u i + 1, 1) = 0 are linear equations and have only one root. □

Appendix B: Proof of the Lemma 4

Proof

Based on Eq. (15), it can be computed inductively that

$$\begin{array}{@{}rcl@{}} \begin{array}{ll} dl^{(i+j)}(x,Y) = &c_{i+j,0}x^{[i]}+ \cdots +{d^{(i)}_{1,i}}^{[-j]}Y^{[i]}+ \cdots \\ &+ {d^{(i)}_{2,0}}^{[-j]}(Y\otimes Y^{[i+j]})^{[0]}+\cdots, \end{array} \end{array}$$
(19)

and

$$\begin{array}{@{}rcl@{}} \begin{array}{ll} dr^{(i+j)}(x,Y) = &c_{i+j,1}x^{[i]}+ \cdots +{d^{(i)}_{1,i}}^{[-j]}Y^{[i]}+ \cdots \\ &+ {d^{(i)}_{2,0}}^{[-j]}(Y\otimes Y^{[i+j]})^{[0]}+\cdots. \end{array} \end{array}$$
(20)

As a result, we have

$$\begin{array}{@{}rcl@{}} \begin{array}{ll} f_{i+j,0}(u_{i+1,0})= c_{i+j,0}+{d^{(i)}_{1,i}}^{[-j]}u_{i+j,0}=0\mbox{, and}\\ f_{i+j,1}(u_{i+1,1})= c_{i+j,1}+{d^{(i)}_{1,i}}^{[-j]}u_{i+j,1}=0. \end{array} \end{array}$$
(21)

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Lin, J., Xie, H. & Yan, Z. Efficient Error Control Decoder Architectures for Noncoherent Random Linear Network Coding. J Sign Process Syst 76, 195–209 (2014). https://doi.org/10.1007/s11265-013-0852-1

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  • DOI: https://doi.org/10.1007/s11265-013-0852-1

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