18,251 research outputs found
On Linear Operator Channels over Finite Fields
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.Comment: 53 pages, 3 figures, submitted to IEEE Transaction on Information
Theor
Capacity Analysis of Linear Operator Channels over Finite Fields
Motivated by communication through a network employing linear network coding,
capacities of linear operator channels (LOCs) with arbitrarily distributed
transfer matrices over finite fields are studied. Both the Shannon capacity
and the subspace coding capacity are analyzed. By establishing
and comparing lower bounds on and upper bounds on , various
necessary conditions and sufficient conditions such that are
obtained. A new class of LOCs such that is identified, which
includes LOCs with uniform-given-rank transfer matrices as special cases. It is
also demonstrated that is strictly less than for a broad
class of LOCs. In general, an optimal subspace coding scheme is difficult to
find because it requires to solve the maximization of a non-concave function.
However, for a LOC with a unique subspace degradation, can be
obtained by solving a convex optimization problem over rank distribution.
Classes of LOCs with a unique subspace degradation are characterized. Since
LOCs with uniform-given-rank transfer matrices have unique subspace
degradations, some existing results on LOCs with uniform-given-rank transfer
matrices are explained from a more general way.Comment: To appear in IEEE Transactions on Information Theor
Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations
This paper establishes information-theoretic limits in estimating a finite
field low-rank matrix given random linear measurements of it. These linear
measurements are obtained by taking inner products of the low-rank matrix with
random sensing matrices. Necessary and sufficient conditions on the number of
measurements required are provided. It is shown that these conditions are sharp
and the minimum-rank decoder is asymptotically optimal. The reliability
function of this decoder is also derived by appealing to de Caen's lower bound
on the probability of a union. The sufficient condition also holds when the
sensing matrices are sparse - a scenario that may be amenable to efficient
decoding. More precisely, it is shown that if the n\times n-sensing matrices
contain, on average, \Omega(nlog n) entries, the number of measurements
required is the same as that when the sensing matrices are dense and contain
entries drawn uniformly at random from the field. Analogies are drawn between
the above results and rank-metric codes in the coding theory literature. In
fact, we are also strongly motivated by understanding when minimum rank
distance decoding of random rank-metric codes succeeds. To this end, we derive
distance properties of equiprobable and sparse rank-metric codes. These
distance properties provide a precise geometric interpretation of the fact that
the sparse ensemble requires as few measurements as the dense one. Finally, we
provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at
IEEE International Symposium on Information Theory (ISIT) 201
Polarization of the Renyi Information Dimension with Applications to Compressed Sensing
In this paper, we show that the Hadamard matrix acts as an extractor over the
reals of the Renyi information dimension (RID), in an analogous way to how it
acts as an extractor of the discrete entropy over finite fields. More
precisely, we prove that the RID of an i.i.d. sequence of mixture random
variables polarizes to the extremal values of 0 and 1 (corresponding to
discrete and continuous distributions) when transformed by a Hadamard matrix.
Further, we prove that the polarization pattern of the RID admits a closed form
expression and follows exactly the Binary Erasure Channel (BEC) polarization
pattern in the discrete setting. We also extend the results from the single- to
the multi-terminal setting, obtaining a Slepian-Wolf counterpart of the RID
polarization. We discuss applications of the RID polarization to Compressed
Sensing of i.i.d. sources. In particular, we use the RID polarization to
construct a family of deterministic -valued sensing matrices for
Compressed Sensing. We run numerical simulations to compare the performance of
the resulting matrices with that of random Gaussian and random Hadamard
matrices. The results indicate that the proposed matrices afford competitive
performances while being explicitly constructed.Comment: 12 pages, 2 figure
Strong converse exponents for the feedback-assisted classical capacity of entanglement-breaking channels
Quantum entanglement can be used in a communication scheme to establish a
correlation between successive channel inputs that is impossible by classical
means. It is known that the classical capacity of quantum channels can be
enhanced by such entangled encoding schemes, but this is not always the case.
In this paper, we prove that a strong converse theorem holds for the classical
capacity of an entanglement-breaking channel even when it is assisted by a
classical feedback link from the receiver to the transmitter. In doing so, we
identify a bound on the strong converse exponent, which determines the
exponentially decaying rate at which the success probability tends to zero, for
a sequence of codes with communication rate exceeding capacity. Proving a
strong converse, along with an achievability theorem, shows that the classical
capacity is a sharp boundary between reliable and unreliable communication
regimes. One of the main tools in our proof is the sandwiched Renyi relative
entropy. The same method of proof is used to derive an exponential bound on the
success probability when communicating over an arbitrary quantum channel
assisted by classical feedback, provided that the transmitter does not use
entangled encoding schemes.Comment: 24 pages, 2 figures, v4: final version accepted for publication in
Problems of Information Transmissio
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