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Rank-Metric Codes and Their Applications
Foundations and Trends® in Communications and Information Theory, 2022 The rank metric measures the distance between two matrices by the rank of their difference. Codes designed for the rank metric have attracted considerable attention in recent years, reinforced by network coding and further motivated by a variety of applications.
Bartz, Hannes +5 more
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Distance metrics for ranked evolutionary trees [PDF]
Proceedings of the National Academy of Sciences, 2020 Significance Rooted binary trees inferred from molecular sequence data provide information about the evolutionary history of populations and species. We introduce metrics on ranked tree shapes and ranked genealogies, in which the shape and temporal branching order in a tree are considered, but not the taxon labels.
Jaehee Kim +2 more
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Bounds on List Decoding of Rank-Metric Codes [PDF]
, 2012 So far, there is no polynomial-time list decoding algorithm (beyond half the minimum distance) for Gabidulin codes. These codes can be seen as the rank-metric equivalent of Reed--Solomon codes.
Wachter-Zeh, Antonia
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Generalized weights: an anticode approach [PDF]
, 2015 In this paper we study generalized weights as an algebraic invariant of a code. We first describe anticodes in the Hamming and in the rank metric, proving in particular that optimal anticodes in the rank metric coincide with Frobenius-closed spaces. Then
Ravagnani, Alberto
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, 2023 Sum-rank metric codes are a natural extension of both linear block codes and rank-metric codes. They have several applications in information theory, including multishot network coding and distributed storage systems. The aim of this chapter is to present the mathematical theory of sum-rank metric codes, paying special attention to the $\mathbb{F}_q ...
Gorla, Elisa +2 more
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Rank metric codes and zeta functions [PDF]
Designs, Codes and Cryptography, 2017 We define the rank-metric zeta function of a code as a generating function of its normalized $q$-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank-metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the ...
I. Blanco-Chacón +3 more
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Generalized Lense-Thirring metrics: higher-curvature corrections and solutions with matter
Journal of High Energy Physics, 2022 The Lense-Thirring spacetime describes a 4-dimensional slowly rotating approximate solution of vacuum Einstein equations valid to a linear order in rotation parameter.
Finnian Gray +4 more
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Hyperbolic rank and subexponential corank of metric spaces [PDF]
, 2001 We introduce a new quasi-isometry invariant $\subcorank X$ of a metric space $X$ called {\it subexponential corank}. A metric space $X$ has subexponential corank $k$ if roughly speaking there exists a continuous map $g:X\to T$ such that for each $t\in T$
Buyalo, Sergei, Schroeder, Viktor
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Tensor Representation of Rank-Metric Codes [PDF]
SIAM Journal on Applied Algebra and Geometry, 2019 We present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We define the tensor rank of a code to be the tensor rank of its generating tensors, and propose that this quantity is a ...
Byrne, Eimear +3 more
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Rank weight hierarchy of some classes of cyclic codes
, 2014 We study the rank weight hierarchy, thus in particular the rank metric, of cyclic codes over the finite field $\mathbb F_{q^m}$, $q$ a prime power, $m \geq 2$.
Ducoat, Jérôme, Oggier, Frédérique
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