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MacWilliams Identity for the Rank Metric [article]

Maximilien Gadouleau, Zhiyuan Yan
2007 arXiv   pre-print
This paper investigates the relationship between the rank weight distribution of a linear code and that of its dual code.  ...  Finally, we determine the relationship between moments of the rank distribution of a linear code and those of its dual code, and provide an alternative derivation of the rank weight distribution of maximum  ...  For the rank metric, it is not clear how we can adapt the concept of complete weight enumerator to give a proof of the MacWilliams identity.  ... 
arXiv:cs/0701097v1 fatcat:jrnxolhdzrhijowvfyebgzf4km

Efficient Encryption from Random Quasi-Cyclic Codes [article]

Carlos Aguilar, Olivier Blazy, Jean-Christophe Deneuville, Philippe Gaborit, Gilles Zémor
2016 arXiv   pre-print
rank metric.  ...  The framework is in the spirit of the schemes first proposed by Alekhnovich in 2003 and based on the difficulty of decoding random linear codes from random errors of low weight.  ...  code of length 2n for rank metric.  ... 
arXiv:1612.05572v1 fatcat:up7iubangna3db2gd77tfi3ywe

Rank-metric codes [article]

Elisa Gorla
2019 arXiv   pre-print
Treated topics include: definition of rank metric, equivalence of codes, support of a codeword and of a code, duality, weight enumerators and MacWilliams identities, higher rank weights, MRD codes, optimal  ...  The chapter gives an introduction to the mathematical theory of rank-metric codes.  ...  We now define the weight distribution, which is an important invariant of a rank-metric code.  ... 
arXiv:1902.02650v1 fatcat:du2pji3eorgc3l5goe5r2hpzja

Codes Endowed With the Rank Metric [article]

Elisa Gorla, Alberto Ravagnani
2017 arXiv   pre-print
We review the main results of the theory of rank-metric codes, with emphasis on their combinatorial properties.  ...  We then investigate the combinatorial structure of MRD codes and optimal anticodes in the rank metric, describing how they relate to each other.  ...  In Section 1 we introduce the most important parameters of a rank-metric code, namely, the minimum distance, the weight distribution, and the distance distribution.  ... 
arXiv:1710.02067v1 fatcat:opocgfsdxbhytawn3lj72q455e

Weighted software metrics aggregation and its application to defect prediction

Maria Ulan, Welf Löwe, Morgan Ericsson, Anna Wingkvist
2021 Empirical Software Engineering  
We propose an automated approach to weighted metrics aggregation that is based on unsupervised learning. It sets metrics scores and their weights based on probability theory and aggregates them.  ...  Manual approaches based on experts do not scale with the number of metrics. Also, experts get confused if the metrics are not independent, which is rarely the case.  ...  Declarations Conflict of Interests The authors declare that they have no conflict of interest.  ... 
doi:10.1007/s10664-021-09984-2 fatcat:cij67aaxgzfulddj3jmqmsobea

Properties of Codes with the Rank Metric [article]

Maximilien Gadouleau, Zhiyuan Yan
2006 arXiv   pre-print
In this paper, we study properties of rank metric codes in general and maximum rank distance (MRD) codes in particular.  ...  For codes with the rank metric, we first establish Gilbert and sphere-packing bounds, and then obtain the asymptotic forms of these two bounds and the Singleton bound.  ...  RANK WEIGHT DISTRIBUTION All public-key cryptosystems based on error codes encrypt the plaintext by first encoding it using the public code and then adding an error vector of weight t (see, for example  ... 
arXiv:cs/0610099v2 fatcat:eidkuwoh3beelecaers2qgoita

MacWilliams Identity for Codes with the Rank Metric

Maximilien Gadouleau, Zhiyuan Yan
2008 EURASIP Journal on Wireless Communications and Networking  
The MacWilliams identity, which relates the weight distribution of a code to the weight distribution of its dual code, is useful in determining the weight distribution of codes.  ...  Using our MacWilliams identity, we also derive related identities for rank metric codes. These identities parallel the binomial and power moment identities derived for codes with the Hamming metric.  ...  ACKNOWLEDGMENTS This work was supported in part by Thales Communications Inc. and in part by a grant from the Commonwealth of Pennsylvania, Department of Community and Economic Development, through the  ... 
doi:10.1155/2008/754021 fatcat:dghizkvxz5doddaj4okwqvvpaa

Matrix Codes as Ideals for Grassmannian Codes and their Weight Properties [article]

Bryan Hernandez, Virgilio Sison
2015 arXiv   pre-print
The rank weight distribution of M_2(GF(q)) is completely determined by the general linear group GL(2,q).  ...  The matrix codes are GF(p)-subspaces of the ring M_2(GF(p)) of 2 × 2 matrices over GF(p) on which the rank metric is applied, and are generated as one-sided proper principal ideals by idempotent elements  ...  Weight properties of rank-metric codes and subspace codes were subsequently examined. The rank weight is not egalitarian nor homogeneous.  ... 
arXiv:1502.05808v1 fatcat:6ci2dbbfvbfm3kp4b4ypq5uuwe

MacWilliams Identity for Codes with the Rank Metric [article]

Maximilien Gadouleau, Zhiyuan Yan
2007 arXiv   pre-print
The MacWilliams identity, which relates the weight distribution of a code to the weight distribution of its dual code, is useful in determining the weight distribution of codes.  ...  Using our MacWilliams identity, we also derive related identities for rank metric codes. These identities parallel the binomial and power moment identities derived for codes with the Hamming metric.  ...  Rank weight distribution of MRD codes The rank weight distribution of linear Class-I MRD codes was given in [4] , [10] .  ... 
arXiv:0706.1751v3 fatcat:ctob2v64qbcehpxlvak7kt7sla

Partition-Balanced Families of Codes and Asymptotic Enumeration in Coding Theory [article]

Eimear Byrne, Alberto Ravagnani
2018 arXiv   pre-print
Finally, we compute the average weight distribution of linear codes in the rank metric, and other parameters that generalize the total weight of a linear code.  ...  Although there does not exist a direct analogue of the redundancy bound for the covering radius of F_q-linear rank metric codes, we show that a similar bound is satisfied by a uniformly random matrix code  ...  Finally, in Section 9 we compute the average weight distributions of Hamming metric, vector rank metric and matrix rank metric codes.  ... 
arXiv:1805.02049v2 fatcat:szqg3cn3n5f67b4qgow23oy5aa

Weight Spectra of Gabidulin Rank-metric Codes and Betti Numbers [article]

Trygve Johnsen, Rakhi Pratihar, Hugues Verdure
2021 arXiv   pre-print
We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid associated to the q-matroid corresponding to the code.  ...  In addition, we demonstrate how the weight distribution and higher weight spectra of such codes can be determined directly from the associated q-matroids by using Möbius functions of its lattice of q-flats  ...  The authors are partially supported by grant 280731 from the Research Council of Norway, and by the project "Pure Mathematics in Norway" through the Trond Mohn Foundation and Tromsø Research Foundation  ... 
arXiv:2106.10993v3 fatcat:rvl6djeniranjeldmrlo7wriry

An Assmus-Mattson Theorem for Rank Metric Codes [article]

Eimear Byrne, Alberto Ravagnani
2019 arXiv   pre-print
We use notions of puncturing and shortening of rank metric codes and the rank-metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold  ...  A collection of matrices in over F_q is said to hold a subspace design if the set of column spaces of its elements forms the blocks of a subspace design.  ...  The weight distributions of a linear matrix code and its dual are related by the rank metric MacWilliams identities [16] . We will use the following formulation from [28, Theorem 31] . Theorem 4.  ... 
arXiv:1806.00448v3 fatcat:li3hcnbpjfbqhj663shfhpfzju

Identity-Based Encryption from Codes with Rank Metric [chapter]

Philippe Gaborit, Adrien Hauteville, Duong Hieu Phan, Jean-Pierre Tillich
2017 Lecture Notes in Computer Science  
We solve this problem by relying on codes with rank metric.  ...  This system, which is an analogue of the McEliece cryptosystem but with a different metric was based on Gabidulin codes, which are analogue codes to Reed-Solomon codes for rank metric.  ...  rank weight in a code that has many codewords of such low rank weight (namely the code C that has been introduced in this section).  ... 
doi:10.1007/978-3-319-63697-9_7 fatcat:wscwpb53mnbq3gvupqhivibf6e

Linear Cutting Blocking Sets and Minimal Codes in the Rank Metric [article]

Gianira N. Alfarano, Martino Borello, Alessandro Neri, Alberto Ravagnani
2021 arXiv   pre-print
The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles.  ...  This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the q-analogues of projective systems and blocking sets.  ...  Acknowledgements The authors of this paper would like to thank John Sheekey and Ferdinando Zullo for fruitful discussions and comments.  ... 
arXiv:2106.12465v1 fatcat:6ck5ndobvzc5lpveylgk673iq4

Rank-Metric Codes and Zeta Functions [article]

I. Blanco-Chacón, E. Byrne, I. Duursma, J. Sheekey
2017 arXiv   pre-print
We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank-metric codes.  ...  We define the rank-metric zeta function of a code as a generating function of its normalized q-binomial moments.  ...  In Section 4 we relate the weight enumerator of a rank-metric code C with its zeta polynomial and obtain the rank-metric analogue of (1.1).  ... 
arXiv:1705.08397v1 fatcat:duq7a45nprchhjcrksshrcgwvi
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