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# Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases

A. Wachter-Zeh's work was supported by the Technical University of Munich|Institute for Advanced Study, funded by the German Excellence Initiative and European Union Seventh Framework Programme under Grant Agreement No. 291763 and the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) unter Grant No. WA3907/1-1

• An interpolation-based decoding scheme for $L$-interleaved subspace codes is presented. The scheme can be used as a (not necessarily polynomial-time) list decoder as well as a polynomial-time probabilistic unique decoder. Both interpretations allow to decode interleaved subspace codes beyond half the minimum subspace distance. Both schemes can decode $\gamma$ insertions and $\delta$ deletions up to $\gamma +L\delta \leq L({{n}_{t}}-k)$, where ${{n}_{t}}$ is the dimension of the transmitted subspace and $k$ is the number of data symbols from the field ${{\mathbb{F}}_{{{q}^{m}}}}$. Further, a complementary decoding approach is presented which corrects $\gamma$ insertions and $\delta$ deletions up to $L\gamma +\delta \leq L({{n}_{t}}-k)$. Both schemes use properties of minimal Gröebner bases for the interpolation module that allow predicting the worst-case list size right after the interpolation step. An efficient procedure for constructing the required minimal Gröebner basis using the general Kötter interpolation is presented. A computationally- and memory-efficient root-finding algorithm for the probabilistic unique decoder is proposed. The overall complexity of the decoding algorithm is at most $\mathcal{O}\left( {{L}^{2}}n_{r}^{2} \right)$ operations in ${{\mathbb{F}}_{{{q}^{m}}}}$ where ${{n}_{r}}$ is the dimension of the received subspace and $L$ is the interleaving order. The analysis as well as the efficient algorithms can also be applied for accelerating the decoding of interleaved Gabidulin codes.

Mathematics Subject Classification: Primary: 94B35, 94B05.

 Citation:

• Figure 1.  Decoding region for the Kötter-Kschischang [18] decoder $(L = 1)$ and for list decoding a homogeneous interleaved KK subspace code with $(L = 4)$. Both codes have minimum subspace distance ${{d}_{s}} = 8$

Figure 2.  Decoding region for Kötter-Kschischang [18] codes $(L = 1)$ and for probabilistic unique decoding of $(L = 4)$-interleaved KK subspace codes. Both codes have minimum subspace distance ${{d}_{s}} = 8$. The decoding region for insertions increases with the interleaving order $L$

Figure 3.  Simulation results and upper bounds on $P_f$ of the probabilistic-unique decoder with parameters $m = {{n}_{t}} = 6, k = 2, L = 2$

Figure 4.  Multiplications vs. the number of insertions for $L = 4$

Figure 5.  Number of multiplications for root-finding step for different interleaving orders L. The complexity of the recursive GE is independent of $\gamma$

Figure 6.  Memory requirements of the root-finding step for ${{n}_{t}} = 80, k = 60$

Table 1.  Simulation results of the probabilistic-unique decoder with parameters $m = {{n}_{t}} = 6, k = 2$ and $L = 2$

 $\gamma + L\delta$ $\gamma$ $\delta$ Transmissions Observed dec. failures Simulated $P_{f}$ 8 8 0 $1.2\cdot10^{6}$ 18749 $1.56\cdot10^{-2}$ 6 1 $4.0\cdot10^{5}$ 6202 $1.55\cdot10^{-2}$ 7 7 0 $4.4\cdot10^{6}$ 1025 $2.33\cdot10^{-4}$ 5 1 $6.0\cdot10^{5}$ 144 $2.40\cdot10^{-4}$ 6 6 0 $9.0\cdot10^{6}$ 21 $2.33\cdot10^{-6}$ 4 1 $3.4\cdot10^{6}$ 17 $5.00\cdot10^{-6}$ 5 5 0 $3.2\cdot10^{6}$ 750 $2.33\cdot10^{-4}$ 3 1 $8.0\cdot10^{5}$ 184 $2.30\cdot10^{-4}$ 4 4 0 $4.4\cdot10^{6}$ 21 $4.77\cdot10^{-6}$ 2 1 $1.6\cdot10^{6}$ 0 $0$

Table 2.  Comparison of different decoding schemes for interleaved KK subspace codes

 Decoding scheme Decoding region Op. in ${{\mathbb{F}}_{{{q}^{m}}}}$ Li-Sidorenko-Silva [20,31] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( Ln_{t}^{2} \right) \phantom{{}^{2}}$ Wachter-Zeh-Zeh [39] $(L+1)t+L\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{2} \right)$ Guruswami-Xing [15] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{6}}n_{t}^{2} \right)$ Bartz-Meier-Sidorenko [4] $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{3}}n_{t}^{3} \right)$ This contribution $(L+1)t+\phantom{L}\varkappa+L\rho\leq L({{n}_{t}}-k)$ $\mathcal{O}\left( {{L}^{4}}n_{t}^{2} \right)$
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