1,553 research outputs found
Saturating systems and the rank covering radius
25 pagesWe introduce the concept of a rank saturating system and outline its correspondence to a rank-metric code with a given covering radius. We consider the problem of finding the value of , which is the minimum -dimension of a -system in which is rank -saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on and evaluate it for certain values of and . We give constructions of rank -saturating systems suggested from geometry
Algebraic number theory and code design for Rayleigh fading channels
Algebraic number theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems.
Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on algebraic number theory are now available.
The aim of this work is to provide both an overview on algebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to algebraic number theory. The basic facts of this mathematical field will be illustrated by many examples and by the use of a computer algebra freeware in order to make it more accessible
to a large audience
Rate-Distortion for Ranking with Incomplete Information
We study the rate-distortion relationship in the set
of permutations endowed with the Kendall t-metric and the
Chebyshev metric. Our study is motivated by the application of permutation rate-distortion to the average-case and worst-case analysis of algorithms for ranking with incomplete information and approximate sorting algorithms. For the Kendall t-metric we provide bounds for small, medium, and large distortion regimes, while for the Chebyshev metric we present bounds that are valid for all distortions and are especially accurate for small
distortions. In addition, for the Chebyshev metric, we provide a construction for covering codes
Fundamental Properties of Sum-Rank Metric Codes
This paper investigates the theory of sum-rank metric codes for which the
individual matrix blocks may have different sizes. Various bounds on the
cardinality of a code are derived, along with their asymptotic extensions. The
duality theory of sum-rank metric codes is also explored, showing that MSRD
codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all
matrix blocks have the same number of columns. In the latter case, duality
considerations lead to an upper bound on the number of blocks for MSRD codes.
The paper also contains various constructions of sum-rank metric codes for
variable block sizes, illustrating the possible behaviours of these objects
with respect to bounds, existence, and duality properties
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