ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 4, pp. 300–320.
Pleiades Publishing, Inc., 2010.
Original Russian Text
E.M. Gabidulin, N.I. Pilipchuk, M. Bossert, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 4, pp. 33–55.
Decoding of Random Network Codes
E. M. Gabidulin
, N. I. Pilipchuk
, and M. Bossert
Moscow Institute of Physics and Technology (State University)
Ulm University, Germany
Received May 14, 2010; in ﬁnal form, September 27, 2010
Abstract—We consider the decoding for Silva–Kschischang–K¨otter random network codes
based on Gabidulin’s rank-metric codes. The model of a random network coding channel can
be reduced to transmitting matrices of a rank code through a channel introducing three types of
additive errors. The ﬁrst type is called random rank errors. To describe other types, the notions
of generalized row erasures and generalized column erasures are introduced. An algorithm for
simultaneous correction of rank errors and generalized erasures is presented. An example is
The lifting construction of network codes based on rank codes in a matrix representation 
is proposed in . The problem of decoding random network codes is reduced to the problem of
decoding rank codes.
A transmitting signal is a code matrix. Its rows are information packets. It is assumed that
an intermediate node, having received a matrix, retransmits linear combinations of its rows. Also,
it might be that some nodes introduce wrong packets, which are added as random linear com-
binations to main messages at the receiver site. Such a model allows one to consider arbitrary
communication networks regardless of their structure and avoid the routing problem.
Due to the code construction and the channel model, one can get extra information about
errors in the received matrix at the receiver site. In this paper, the notions of generalized row
and columns erasures are introduced. Ranks of erasures, as well as extra information about errors,
are determined before decoding. Erasure correction requires error capacity twice as small as when
correcting random rank errors. In fact, extra information derived due to the lifting construction of
a code matrix increases the correcting capability of a rank code.
Algorithms for simultaneous correction of random rank errors and speciﬁc erasures were de-
scribed in [1,3,4]. Here we present an extension of the algorithm from  to the case of generalized
erasures. Conditions are given for various combinations of errors and erasures to appear, in partic-
ular, for the case where only erasures exist and error correction is unnecessary .
2. RANK METRIC AND RANK CODES
The following notation is used below: F
is a ﬁnite ﬁeld of q elements, which is called in what
follows the ground ﬁeld; F
is its extension of degree n; F
is the space of n × m matrices
is the m-dimensional space of vectors over F
Supported in part by the Analytical Departmental Target Program “Development of the Scientiﬁc Poten-
tial of the Higher School (2009–1010),” project no. 2391.