Generalized Fano and non-Fano networks

@article{Das2016GeneralizedFA,
  title={Generalized Fano and non-Fano networks},
  author={Niladri Das and Brijesh Kumar Rai},
  journal={2017 Twenty-third National Conference on Communications (NCC)},
  year={2016},
  pages={1-6}
}
  • Niladri DasB. K. Rai
  • Published 19 September 2016
  • Computer Science, Mathematics
  • 2017 Twenty-third National Conference on Communications (NCC)
It is known that the Fano network has a vector linear solution if and only if the characteristic of the finite field is 2; and the non-Fano network has a vector linear solution if and only if the characteristic of the finite field is not 2. Using these properties of Fano and non-Fano networks it has been shown that linear network coding is insufficient. In this paper we generalize the properties of Fano and non-Fano networks. Specifically, by adding more nodes and edges to the Fano network, we… 

Figures from this paper

On fractional linear network coding solution of multiple-unicast networks

  • Niladri DasB. K. Rai
  • Mathematics, Computer Science
    2017 Twenty-third National Conference on Communications (NCC)
  • 2017
It is shown that for any non-zero positive rational number k/n, there exists a multiple-unicast network which has a rate k/N fractional linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes.

Generalized Fano and non-Fano networks

  • Niladri DasB. K. Rai
  • Computer Science, Mathematics
    2017 Twenty-third National Conference on Communications (NCC)
  • 2017
This paper generalizes the properties of Fano and non-Fano networks and constructs a network which has a vector linear solution for any vector dimension if and only if the characteristic of the finite field belongs to an arbitrary given set of primes.

References

SHOWING 1-8 OF 8 REFERENCES

Networks, Matroids, and Non-Shannon Information Inequalities

The Vamos network is constructed, and it is proved that Shannon-type information inequalities are insufficient even for computing network coding capacities of multiple-unicast networks.

On Network Coding for Sum-Networks

  • B. K. RaiB. Dey
  • Computer Science, Mathematics
    IEEE Transactions on Information Theory
  • 2012
The insufficiency of linear net- work coding and unachievability of the network coding capacity are proved for sum-networks by using similar known results for communication networks.

Some results on communicating the sum of sources over a network

For any finite set of primes, there exists a network where the sum of the sources can be communicated to the terminals only over finite fields of characteristic belonging to that set.

Scalar-linear solvability of matroidal networks associated with representable matroids

  • A. KimM. Médard
  • Mathematics, Computer Science
    2010 6th International Symposium on Turbo Codes & Iterative Information Processing
  • 2010
It is shown that determining scalar-linear solvability of a network is equivalent to finding a representable matroid over a finite field and a valid network-matroid mapping and that this result, combined with the construction method due to Dougherty et al., can generate potentially new Scalar-linearly solvable networks.

On Linear Network Coding

This work demonstrates how codes with one notion of linearity can be used to build, in a distributed manner, codes with another notion oflinearity, and introduces the new class of filter-bank network codes of which all previous definitions of linear network codes are special cases.

Achievable Rate Regions for Network Coding

In addition to the known matrix-computation method for proving outer bounds for linear coding, a new method is presented that yields actual characteristic-dependent linear rank inequalities from which the desired bounds follow immediately.

Insufficiency of linear coding in network information flow

It is shown that the network coding capacity of this counterexample network is strictly greater than the maximum linear coding capacity over any finite field, so the network is not even asymptotically linearly solvable.

Generalized Fano and non-Fano networks

  • Niladri DasB. K. Rai
  • Computer Science, Mathematics
    2017 Twenty-third National Conference on Communications (NCC)
  • 2017
This paper generalizes the properties of Fano and non-Fano networks and constructs a network which has a vector linear solution for any vector dimension if and only if the characteristic of the finite field belongs to an arbitrary given set of primes.