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 Y = AX + BE, where X is the channel input, Y is the channel output, E is random error, and A and B are random transfer matrices. Tight capacity results are obtained and simple polynomial-complexity capacity-achieving coding schemes are provided under the assumption that A is uniform over all full-rank matrices and BE is uniform over all rank-t matrices, extending the work of Silva, Kschischang, and Kötter (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.

Though network coding is traditionally performed over finite fields, recent work on nested-lattice-based network coding suggests that, by allowing network coding over finite rings, more efficient physical-layer network coding schemes can be constructed. This paper considers the problem of communication over a finite-chain-ring matrix channel Y = AX + BZ, where X is the channel input, Y is the channel output, Z is random noise, and A and B are random transfer matrices. Tight capacity results are obtained and simple polynomial-complexity capacity-achieving coding schemes are provided under certain distributions of A, B, and Z, extending the work of Silva, Kschischang and Kötter (2010), who handled the case of finite fields. This extension is based on several new results that generalize concepts and methods from matrices over finite fields to matrices over finite chain rings.

This paper deals with the multiplicative finite-field matrix channel, a discrete memoryless channel whose input and output are matrices (over a finite field) related by a multiplicative transfer matrix. The model considered here assumes that all transfer matrices with the same rank are equiprobable, so that the channel is completely characterized by the rank distribution of the transfer matrix. This model is seen to be more flexible than previously proposed ones in describing random linear network coding systems subject to link erasures, while still being sufficiently simple to allow tractability. The model is also conservative in the sense that its capacity provides a lower bound on the capacity of any channel with the same rank distribution. A main contribution is to express the channel capacity as the solution of a convex optimization problem which can be easily solved by numerical computation. For the special case of constant-rank input, a closed-form expression for the capacity is obtained. The behavior of the channel for asymptotically large field size or packet length is studied, and it is shown that constant-rank input suffices in this case. Finally, it is proved that the well-known approach of treating inputs and outputs as subspaces is information-lossless even in this more general model.

This paper deals with the multiplicative finite-field matrix channel, a discrete memoryless channel whose input and output are matrices (over a finite field) related by a multiplicative transfer matrix. Our model allows this transfer matrix to have any rank, while assuming that all transfer matrices with the same rank are equiprobable. While in general the capacity cannot be obtained in closed form, we provide a simplification of the problem (from qnmto O(n) variables) which allows for easy numerical computation. A tight upper bound on the capacity is also derived, and for the special case of constant-rank input, we obtain an exact formula. Several existing results can be obtained as special cases of our approach. In addition, we prove that the well-known approach of treating inputs and outputs as subspaces is information-lossless even in this more general case.

The subspace channel was introduced by Koetter and Kschischang as an adequate model for the communication channel from the source node to a sink node of a multicast network that performs random linear network coding. So far, attention has been given to one-shot subspace codes, that is, codes that use the subspace channel only once. In contrast, this paper explores the idea of using the subspace channel more than once and investigates the so called multishot subspace codes. We present definitions for the problem, a motivating example, lower and upper bounds for the size of codes, and a multilevel construction of codes based on block-coded modulation.