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# Multishot Codes for Network Coding using Rank-Metric Codes 

Abstract —The multiplicative-additive finite-field matrix chan- nel arises as an adequate model for linear network coding systems when links are subject to errors and erasures, and both the network topology and the network code are unknown. In a previous work we proposed a general construction of multishot codes for this channel based on the multilevel coding theory. Herein we apply this construction to the rank-metric space, obtaining multishot rank-metric codes which, by lifting, can be converted to codes for the aforementioned channel. We also adapt well-known encoding and decoding algorithms to the considered situation. 

Noncoherent linear network coding with unreliable links in a multicast scenario has been given a good deal of attention since the seminal work of Koetter and Kschischang [1]. The problem is suitably modeled by the multiplicative-additive finite-field matrix channel given by 

where $^{X}$is the N$_{i}$ -by- T channel input matrix, $^{Y}$is the N$_{o}$ - by- T channel output matrix, $^{A}$is the N$_{o}$ -by- N$_{i}$ multiplicative transfer matrix, and $^{Z}$is the N$_{o}$ -by- T additive error matrix. All matrices are over some finite field F$_{q}$ . 

In the context of network coding, N$_{i}$ stands for the number of packets transmitted by the source node while N$_{o}$ stands for the number of packets received by a given sink node at each time slot, a packet consisting of T symbols from F$_{q}$ ; the matrices $^{X}$and $^{Y}$are then formed by juxtaposing the N$_{i}$ transmitted and the N$_{o}$ received packets, respectively, seen as row vectors. Matrix $^{A}$represents the network transfer matrix, which depends on the network topology and network code used (both assumed unknown in a noncoherent scenario), and matrix $^{Z}$is related to link errors $^{1}$. For simplicity, we set N$_{i}$ = N$_{o}$ = N throughout this work. 

To achieve a reliable communication over this channel, matrix codes are employed. So far, attention has mostly been given to one-shot matrix codes, that is, codes that use the matrix channel (1) only once. In this case, a code is simply a non-empty subset of matrices. Among the existing 

## I. INTRODUCTION 

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Roberto W. Nóbrega and Bartolomeu F. Uchôa-Filho Communications Research Group Department of Electrical Engineering Federal University of Santa Catarina Florianópolis, SC, 88040-900, Brazil {rwnobrega, uchoa}@eel.ufsc.br 

constructions for one-shot codes, we highlight that of Silva et al. [2], in which matrix codes are obtained from rank-metric codes . Rank-metric codes, in turn, were already studied before (e.g., by Gabidulin [3] and Roth [4]) in distinct contexts. 

In contrast, we herein consider blocks of n consecutive uses of the matrix channel (1). Under this framework, a code is now a non-empty subset of n -tuples of matrices; we call it an n - shot (or multishot ) matrix code. Multishot codes for network coding were studied in [5], where a general construction based on the well-known block coded modulation and multilevel code construction of Imai and Hirakawa [6] has been proposed. It has been shown in [5] that multishot codes can correct more errors than one-shot codes, which motivates this work. 

In the present paper, we combine the rank-metric approach of Silva et al. with the multilevel code construction of Imai and Hirakawa to obtain multishot codes for network coding. The problem can also be interpreted within the theory of generalized concatenated codes of Blokh and Zyablov [7], and by doing so it is possible to adapt well-known encoding and decoding procedures to the case at hand (cf. [8, Chapter 15]). 

We begin in Section II by providing background on con- cepts from one-shot noncoherent network coding with errors, including an overview of rank-metric codes applied to network coding. In Section III we introduce multishot matrix codes. Section IV presents a brief review of the general multilevel theory to construct block codes over arbitrary metric spaces. 

The main contribution is Section V, where we particularize the multilevel theory to the rank-metric space, derive an encoding procedure based on coset partitioning, and adapt a hard-decision multistage decoding algorithm to our problem. Finally, Section VI concludes the paper. 

## A. Error Model 

This work follows the approach of Silva, Kschischang, and Koetter [2], in which the adversities of the matrix channel (1) come in two flavors: the rank-deficiency of the multiplicativ e transfer matrix and the rank of the additive error matrix. These are constrained to 

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## II. BACKGROUND 

