37,326 research outputs found
A characterization of MDS codes that have an error correcting pair
Error-correcting pairs were introduced in 1988 by R. Pellikaan, and were
found independently by R. K\"otter (1992), as a general algebraic method of
decoding linear codes. These pairs exist for several classes of codes. However
little or no study has been made for characterizing those codes. This article
is an attempt to fill the vacuum left by the literature concerning this
subject. Since every linear code is contained in an MDS code of the same
minimum distance over some finite field extension we have focused our study on
the class of MDS codes.
Our main result states that an MDS code of minimum distance has a
-ECP if and only if it is a generalized Reed-Solomon code. A second proof is
given using recent results Mirandola and Z\'emor (2015) on the Schur product of
codes
Relative Generalized Rank Weight of Linear Codes and Its Applications to Network Coding
By extending the notion of minimum rank distance, this paper introduces two
new relative code parameters of a linear code C_1 of length n over a field
extension and its subcode C_2. One is called the relative
dimension/intersection profile (RDIP), and the other is called the relative
generalized rank weight (RGRW). We clarify their basic properties and the
relation between the RGRW and the minimum rank distance. As applications of the
RDIP and the RGRW, the security performance and the error correction capability
of secure network coding, guaranteed independently of the underlying network
code, are analyzed and clarified. We propose a construction of secure network
coding scheme, and analyze its security performance and error correction
capability as an example of applications of the RDIP and the RGRW. Silva and
Kschischang showed the existence of a secure network coding in which no part of
the secret message is revealed to the adversary even if any dim C_1-1 links are
wiretapped, which is guaranteed over any underlying network code. However, the
explicit construction of such a scheme remained an open problem. Our new
construction is just one instance of secure network coding that solves this
open problem.Comment: IEEEtran.cls, 25 pages, no figure, accepted for publication in IEEE
Transactions on Information Theor
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