On Network Coding for Funnel Networks

Karimian, Pourya; Borujeny, Reza Rafie; Ardakani, Masoud

doi:10.1109/lcomm.2015.2477816

Cited by 1 publication

(1 citation statement)

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“…al [18] presented polynomial-time algorithms for designing scalar linear codes for multicast networks. Karimian, Borujeny, and Ardakani [19] showed there exists a class of nonmulticast networks for which random scalar linear coding algorithms fail with high probability and presented a new approach to random scalar linear network coding for such networks. Rasala Lehman and Lehman [26] and Tavory, Feder, and Ron [31] independently showed that some solvable multicast networks asymptotically require finite field alphabets to be at least as large as twice the square root of the number of receiver nodes in order to achieve scalar linear solutions.…”

Section: Related Workmentioning

confidence: 99%

Linear Network Coding Over Rings – Part I: Scalar Codes and Commutative Alphabets

Connelly

Zeger

2018

IEEE Trans. Inform. Theory

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Fixed-size commutative rings are quasi-ordered such that all scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher-ordered ring. As consequences, if a network has a scalar linear solution over some finite commutative ring, then (i) the network is also scalar linearly solvable over a maximal commutative ring of the same size, and (ii) the (unique) smallest size commutative ring over which the network has a scalar linear solution is a field. We prove that a commutative ring is maximal with respect to the quasi-order if and only if some network is scalar linearly solvable over the ring but not over any other commutative ring of the same size. Furthermore, we show that maximal commutative rings are direct products of certain fields specified by the integer partitions of the prime factor multiplicities of the maximal ring's size.Finally, we prove that there is a unique maximal commutative ring of size m if and only if each prime factor of m has multiplicity in {1, 2, 3, 4, 6}. In fact, whenever p is prime and k ∈ {1, 2, 3, 4, 6}, the unique such maximal ring of size p k is the field GF p k . However, for every field GF p k with k ∈ {1, 2, 3, 4, 6}, there is always some network that is not scalar linearly solvable over the field but is scalar linearly solvable over a commutative ring of the same size. These results imply that for scalar linear network coding over commutative rings, fields can always be used when the alphabet size is flexible, but alternative rings may be needed when the alphabet size is fixed.

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Section: Related Workmentioning

confidence: 99%