20,215 research outputs found
Communication over Finite-Chain-Ring Matrix Channels
Though network coding is traditionally performed over finite fields, recent
work on nested-lattice-based network coding suggests that, by allowing network
coding over certain finite rings, more efficient physical-layer network coding
schemes can be constructed. This paper considers the problem of communication
over a finite-ring matrix channel , where is the channel
input, is the channel output, is random error, and and are
random transfer matrices. Tight capacity results are obtained and simple
polynomial-complexity capacity-achieving coding schemes are provided under the
assumption that is uniform over all full-rank matrices and is uniform
over all rank- matrices, extending the work of Silva, Kschischang and
K\"{o}tter (2010), who handled the case of finite fields. This extension is
based on several new results, which may be of independent interest, that
generalize concepts and methods from matrices over finite fields to matrices
over finite chain rings.Comment: Submitted to IEEE Transactions on Information Theory, April 2013.
Revised version submitted in Feb. 2014. Final version submitted in June 201
Efficient Computation of the Characteristic Polynomial
This article deals with the computation of the characteristic polynomial of
dense matrices over small finite fields and over the integers. We first present
two algorithms for the finite fields: one is based on Krylov iterates and
Gaussian elimination. We compare it to an improvement of the second algorithm
of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third
algorithm could improve both complexity and computational time. We use these
results as a basis for the computation of the characteristic polynomial of
integer matrices. We first use early termination and Chinese remaindering for
dense matrices. Then a probabilistic approach, based on integer minimal
polynomial and Hensel factorization, is particularly well suited to sparse
and/or structured matrices
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