( r ( i ) j )$_{0}$$_{≤}$$_{j<n}$ ( L ′ $_{j}$, E ′ $_{j}$)$_{0}$$_{≤}$$_{j<n}$ (˜ u ( i ) j )$_{0}$$_{≤}$$_{j<n}$ (˜ v ( i ) j )$_{0}$$_{≤}$$_{j<n}$ ( ˜ m ( i ) j )$_{0}$$_{≤}$$_{j<n}$ ≡ ˜ c ( i ) ˆ c ( i ) Dec R$_{i}$ Dec H$_{i}$ Coset Leader Enc − 1 [ R$_{i}$ / R$_{i}$$_{+1}$ ] Figure 3. Block diagram for concatenated decoder at i -th stage. ( r (0) j )$_{0}$$_{≤}$$_{j<n}$ ( r (1) j )$_{0}$$_{≤}$$_{j<n}$ ( r ( m − 1) j )$_{0}$$_{≤}$$_{j<n}$ . − − ˆ c (0) ˆ c (1) ˆ c ( m − 1) ConcatDec 0 ConcatDec 1 ConcatDec m − 1 Enc [ R$_{0}$ / R$_{1}$ ] Enc [ R$_{1}$ / R$_{2}$ ] (ˆ v (0) j )$_{0}$$_{≤}$$_{j<n}$ (ˆ v (1) j )$_{0}$$_{≤}$$_{j<n}$ Figure 4. Block diagram for hard-decision multistage decoder. Each one of the ConcatDec blocks is detailed in Figure 3. ( r $_{j}$)$_{0}$$_{≤}$$_{j<n}$ We now suggest a sub-optimal hard-decision multistage decoding algorithm [8, Section 15.2] for the case when all R$_{i}$ are Gabidulin codes. Let X ⊆ ( F N × T q ) n be a multishot matrix code obtained by the multilevel construction just described. Following the model in Section III, let ( X $_{0}$, . . . , X n − $_{1}$) ∈ X be the transmitted codeword and ( Y $_{0}$, . . . , Y n − $_{1}$) ∈ ( F N × T q ) n be the received sequence. ( r (2) j )$_{0}$$_{≤}$$_{j<n}$ D. Hard-Decision Multistage Decoding The multistage decoding occurs in m stages; we start by finding the reduction ( r $_{j}$, L ′ $_{j}$, E ′ $_{j}$) ∈ F N q M × F N × µ q × F δ × M q of each Y j ∈ F T × M q and setting r (0) j = $^{r}$$_{j}$. The decoding then proceeds in an iterative fashion. For 0 ≤ i < m , at the i -th stage, the following steps are executed (Figure 3). 1) Inner decoding . Using the generalized decoding method of Silva etal. [2], decode ( r ( i ) j , L ′ $_{j}$, E ′ $_{j}$) into a codeword ˜ u ( i ) j ∈ R$_{i}$ . Of course ˜ u ( i ) j belongs to some coset in R$_{i}$ / R$_{i}$$_{+1}$ . Identify the leader ˜ v ( i ) j ∈ [ R$_{i}$ / R$_{i}$$_{+1}$ ] of this coset and find—by inverse mapping—the message ˜ m ( i ) j ∈ F K i − K $^{i}$+1 q M that generated it. 2) After n inner decodings we obtain ( ˜ m ( i ) 0 , . . . , ˜ m ( i ) n − $_{1}$) . Similarly to the encoding procedure, define ˜ c ( i ) as the vector in F n p i corresponding to ( ˜ m ( i ) 0 , . . . , ˜ m ( i ) n − $_{1}$) . 3) Outer decoding . Note that, due to errors, ˜ c ( i ) may not be a codeword of H$_{i}$ . Therefore, decode ˜ c ( i ) into the closest (in Hamming sense) codeword ˆ c ( i ) ∈ H$_{i}$ . [1] R. Koetter and F. R. Kschischang, “Coding for errors and e rasures in random network coding,” IEEE Transactions on Information Theory , vol. 54, pp. 3579–3591, Aug. 2008. [2] D. Silva, F. R. Kschischang, and R. Koetter, “A rank-metr ic approach to error control in random network coding,” IEEE Transactions on Information Theory , vol. 54, pp. 3951–3967, Sept. 2008. [3] E. M. Gabidulin, “Theory of codes with maximum rank dista nce,” Problemy Peredachi Informatsii , vol. 21, no. 1, pp. 3–16, 1985. ACKNOWLEDGMENT REFERENCES VI. CONCLUSION The authors would like to thank Danilo Silva for helpful discussions, and CAPES (Brazil) and CNPq (Brazil) for bib- liographical research and financial support. At the end of the iteration, each r ( i ) j is updated to where each ˆ v ( i ) j is obtained from ˆ c ( i ) according to [ R$_{i}$ / R$_{i}$$_{+1}$ ] (just like in Steps 2 and 3 of the encoding procedure) and decoding proceeds to the next stage. After m steps, we have ˆ c $^{(0)}$, . . . , ˆ c ( m − $^{1)}$. Figure 4 illustrates the whole process. r ( i +1) j = r ( i ) j − ˆ v ( i ) j , This work presented a bit more explicit multilevel construc- tion of multishot codes for network coding than that introduced in [5]. A natural question arises: how good are the proposed codes? This demands a comparison of (9) with known bounds or previous one-shot constructions. Another open problem is to adapt soft-decision multistage decoding algorithms [8, Section 15.3] to the current scenario. In particular, the metric is now the rank distance (as opposed to the Euclidean distance in the case of codes for the AWGN channel). This can possibly take advantage of list decoding of rank-metric codes. [4] R. M. Roth, “Maximum-rank array codes and their applicat ion to criss- cross error correction,” IEEE Transactions on Information Theory , vol. 37, pp. 328–336, Mar. 1991. [5] R. W. Nóbrega and B. F. Uchôa-Filho, “Multishot codes for network coding: Bounds and a multilevel construction,” in Proceedings of the 2009 IEEE International Symposium on Information Theory (ISIT’09) , (Seoul, South Korea), June 2009. [6] H. Imai and S. Hirakawa, “A new multilevel coding method u sing error- correcting codes,” IEEE Transactions on Information Theory , vol. 23, pp. 371–377, May 1977. [7] E. L. Blokh and V. V. Zyablov, “Coding of generalized conc atenated codes,” Problemy Peredachi Informatsii , vol. 10, no. 3, pp. 45–50, 1974. [8] S. Lin and D. J. Costello Jr., Error Control Coding . Pearson Prentice Hall, 2nd ed., 2004. [9] D. Silva and F. R. Kschischang, “On metrics for error corr ection in network coding,” IEEE Transactions on Information Theory , vol. 55, pp. 5479–5490, Dec. 2009.