The capacity of a finite field matrix channel
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The Additive-Multiplicative Matrix Channel (AMMC) was introduced by Silva, Kschischang and Kötter in 2010 to model data transmission using random linear network coding. The input and output of the channel are n× m matrices over a finite field 𝔽_q. On input the matrix X, the channel outputs Y=A(X+W) where A is a uniformly chosen n× n invertible matrix over 𝔽_q and where W is a uniformly chosen n× m matrix over 𝔽_q of rank t. Silva et al considered the case when 2n≤ m. They determined the asymptotic capacity of the AMMC when t, n and m are fixed and q→∞. They also determined the leading term of the capacity when q is fixed, and t, n and m grow linearly. We generalise these results, showing that the condition 2n≥ m can be removed. (Our formula for the capacity falls into two cases, one of which generalises the 2n≥ m case.) We also improve the error term in the case when q is fixed.
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