ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 4, pp. 343–356.
Pleiades Publishing, Inc., 2009.
Original Russian Text
E.M. Gabidulin, M. Bossert, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 4, pp. 54–68.
Algebraic Codes for Network Coding
E. M. Gabidulin
and M. Bossert
Moscow Institute of Physics and Technology (State University)
Ulm University, Germany
Received January 26, 2009; in ﬁnal form, October 1, 2009
Abstract—The subspace metric is an object of active research in network coding. Nevertheless,
little is known on codes over this metric. In the present paper, several classes of codes over
subspace metric are deﬁned and investigated, including codes with distance 2, codes with the
maximal distance, and constant-distance constant-dimension codes. Also, Gilbert-type bounds
Network coding is a relatively new area. It is based on a slightly modiﬁed simple mathematical
model of information ﬂows and communication in ordinary networks. Traditionally, intermediate
nodes in a network are allowed to store and forward information packets. In a pioneering paper 
the notion of network coding was introduced. A node is allowed to create a linear combination of
several received packets and transmit this combination instead of transmitting each packet sepa-
rately. It was shown by an example that the throughput can be increased. The coding gain, i.e.,
the ratio of the throughput using network coding to that without network coding, was intensively
studied (see, e.g., [2–5]). Afterwards, transmitting several linear combinations together with the
corresponding coeﬃcients was proposed. Linear combinations can be either random  or determin-
istic . Some other results based on network coding were established in various areas of computer
science. The problem of data security was studied in [8–11]. Combination of network coding with
broadcasting in wireless networks was investigated in [12, 13].
Network coding has also many interconnections with error-correcting codes [14–18] and infor-
mation theory .
Usually, speciﬁc network conﬁgurations are required for applying network codes. Hence, network
coding becomes “network-conﬁguration-dependent.” This is inconvenient from both practical and
theoretical points of view.
A new (the so-called subspace) approach to network coding was proposed in . It makes it
possible to bypass many previous restrictions on network conﬁgurations. Coding schemes using this
approach were developed in . The subspace approach is based on a subspace metric deﬁned in
the next section. Given this metric, there arise standard problems of algebraic coding: construct-
ing codes in this metric, deriving bounds for optimal codes, developing encoding and decoding
This paper is mainly devoted to constructing codes with a prescribed code distance.
We also consider Gilbert-type bounds for codes in the subspace metric. A weak bound, proposed
in , was based on a wrong statement; however, the bound itself is valid. A strong bound was
ﬁrst presented in .