Generalized Fano and non-Fano networks

Niladri Das; Brijesh Kumar Rai

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arXiv · 1609.05815

Generalized Fano and non-Fano networks

Niladri Das·Brijesh Kumar Rai

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Abstract

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 construct 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 $\{p_1,p_2,\ldots,p_l\}$. Similarly, by adding more nodes and edges to the non-Fano network, we construct a network which has a vector linear solution for any vector dimension if and only if the characteristic of the finite field does not belong to an arbitrary given set of primes $\{p_1,p_2,\ldots,p_l\}$.

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Niladri Das, Brijesh Kumar Rai. 2016-09-19. Generalized Fano and non-Fano networks. https://arxiv.org/abs/1609.05815

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