On the Message Dimensions of Vector Linearly Solvable Networks

It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is 2. It is not known whether this result can be generalized for an arbitrary message dimension. In this letter, we show that there exist...

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Published in:IEEE communications letters Vol. 20; no. 9; pp. 1701 - 1704
Main Authors: Das, Niladri, Rai, Brijesh Kumar
Format: Journal Article
Language:English
Published: New York IEEE 01.09.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
discrete polymatroids
Electronic mail
Indexes
Linear codes
Matroid theory
Network coding
Routing
Scalar linear network coding
vector linear network coding
Zinc
ISSN:1089-7798, 1558-2558
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Abstract It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is 2. It is not known whether this result can be generalized for an arbitrary message dimension. In this letter, we show that there exists a network that admits an m dimensional vector linear solution, where m is a positive integer greater than or equal to 2, but does not have a vector linear solution over any finite field when the message dimension is less than m.
AbstractList It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is 2. It is not known whether this result can be generalized for an arbitrary message dimension. In this letter, we show that there exists a network that admits an [Formula Omitted] dimensional vector linear solution, where [Formula Omitted] is a positive integer greater than or equal to 2, but does not have a vector linear solution over any finite field when the message dimension is less than [Formula Omitted].
It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is 2. It is not known whether this result can be generalized for an arbitrary message dimension. In this letter, we show that there exists a network that admits an m dimensional vector linear solution, where m is a positive integer greater than or equal to 2, but does not have a vector linear solution over any finite field when the message dimension is less than m.
Author Das, Niladri
Rai, Brijesh Kumar
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10.1023/A:1021852421716
10.1109/TCOMM.2016.2613085
10.1109/ISTC.2010.5613864
10.1109/18.850663
10.1109/TIT.2005.851744
10.1109/TIT.2002.807285
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SubjectTerms discrete polymatroids
Electronic mail
Indexes
Linear codes
Matroid theory
Network coding
Routing
Scalar linear network coding
vector linear network coding
Zinc
Title On the Message Dimensions of Vector Linearly Solvable Networks
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