Coding for linear operator channels over finite fields

Abstract

Motivated by linear network coding, communication channels perform linear operation over finite fields, namely linear operator channels (LOCs), are studied in this paper. For such a channel, its output vector is a linear transform of its input vector, and the transformation matrix is randomly and independently generated. The transformation matrix is assumed to remain constant for every T input vectors and to be unknown to both the transmitter and the receiver.There are NO constraints on the distribution of the transformation matrix and the field size.Specifically, the optimality of subspace coding over LOCs is investigated. A lower bound on the maximum achievable rate of subspace coding is obtained and it is shown to be tight for some cases. The maximum achievable rate of constant-dimensional subspace coding is characterized and the loss of rate incurred by using constant-dimensional subspace coding is insignificant.The maximum achievable rate of channel training is close to the lower bound on the maximum achievable rate of subspace coding. Two coding approaches based on channel training are proposed and their performances are evaluated.Our first approach makes use of rank-metric codes and its optimality depends on the existence of maximum rank distance codes. Our second approach applies linear coding and it can achieve the maximum achievable rate of channel training. Our code designs require only the knowledge of the expectation of the rank of the transformation matrix.The second scheme can also be realized ratelessly without a priori knowledge of the channel statistics.

On Multiplicative Matrix Channels over Finite Chain Rings

“…Finally, we adapt the coding scheme to the noncoherent scenario, in which the instances of the transfer matrix are unknown to both the transmitter and receiver. Our results extend (and make use of) some of those obtained by Yang et al in [11]- [13] and Silva et al in [15], [16], which address the finite field case. It is also worth mentioning that a generalization of the results in [10] from finite fields to finite chain rings is presented in [17].…”

“…In this case, we compute invertible matrices P and Q such that A = P DQ, where D is the Smith normal form of A, as given by (11). We then setỸ P −1 Y andX QX, so that we can communicate using the equivalent channelỸ = DX by employing the same scheme as before.…”

“…Also in [11], [12], two multi-shot coding schemes for MMCs over finite fields are proposed, which are able to achieve the channel capacity given in Theorem 1. The first scheme makes use of rank-distance codes (more on these later) and requires ≥ n in order to be capacity-achieving; the second scheme is based on random coding and imposes no restriction on .…”

“…Recall that the two coding schemes proposed in [11], [12] (see Section V) are universal and have polynomial time complexity. Consequently, by using them as component codes, we can obtain a universal, capacity-achieving composite code with polynomial time complexity.…”

“…Finite-field MMCs have been studied under an informationtheoretic approach according to different assumptions on the probability distribution of the transfer matrix [9]- [14]. In this work, following parts of [11], [12], we consider finitechain-ring MMCs under a coherent scenario, meaning that we assume that the instances of the transfer matrix A are available to the receiver (but unknown to the transmitter). Besides that, we impose no restrictions on the statistics of A, except that A must be independent of X.…”