Linear Network Coding: Effects of Varying the Message Dimension on the Set of Characteristics
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It is known a vector linear solution may exist if and only if the characteristic of the finite field belongs to a certain set of primes. But, can increasing the message dimension make a network vector linearly solvable over a larger set of characteristics? To the best of our knowledge, there exists no network in the literature which has a vector linear solution for some message dimension if and only if the characteristic of the finite field belongs to a set P, and for some other message dimension it has a vector linear solution over some finite field whose characteristic does not belong to P. We have found that by increasing the message dimension just by 1, the set of characteristics over which a vector linear solution exists may get arbitrarily larger. However, somewhat surprisingly, we have also found that by decreasing the message dimension just by 1, the set of characteristics over which a vector linear solution exists may get arbitrarily larger. As a consequence of these finding, we prove two more results: (i) rings may be superior to finite fields in terms of achieving a scalar linear solution over a lesser sized alphabet, (ii) existences of m_1 and m_2 dimensional vector linear solutions guarantees the existence of an (m_1 + m_2)-dimensional vector linear solution only if the m_1 and m_2 dimensional vector linear solutions exist over the same finite field.
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