On Vector Linear Solvability of Multicast Networks

Vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an L-dimensional vector of data symbols over a base field GF(q). Vector LNC enriches the choices of coding operations at intermediate nodes, and there is a popu...

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Bibliographic Details
Published in:IEEE transactions on communications 2016-12, Vol.64 (12), p.5096-5107
Main Authors: Sun, Qifu Tyler, Xiaolong Yang, Keping Long, Xunrui Yin, Zongpeng Li
Format: Article
Language:English
Subjects:
alphabet size
Coding
Data models
direct sum
Linear codes
Mathematical model
Multicast
multicast networks
Network coding
Networks
Receivers
scalar network coding
Vector network coding
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Summary:Vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an L-dimensional vector of data symbols over a base field GF(q). Vector LNC enriches the choices of coding operations at intermediate nodes, and there is a popular conjecture on the benefit of vector LNC over scalar LNC in terms of alphabet size of data units: there exist (singlesource) multicast networks that are vector linearly solvable of dimension L over GF(q) but not scalar linearly solvable over any field of size q' qL. This paper introduces a systematic way to construct such multicast networks, and subsequently establish explicit instances to affirm the positive answer of this conjecture for infinitely many alphabet sizes pL with respect to an arbitrary prime p. On the other hand, this paper also presents explicit instances with the special property that they do not have a vector linear solution of dimension L over GF(2) but have scalar linear solutions over GF(q') for someq' L.
ISSN:0090-6778
1558-0857
DOI:10.1109/TCOMM.2016.2613085