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On conjugacy classes of subgroups of the general linear group and cyclic orbit codes

2011
*
2011 IEEE International Symposium on Information Theory Proceedings
*

We start by classifying

doi:10.1109/isit.2011.6033885
dblp:conf/isit/ManganielloTR11
fatcat:q67oir42yngonfi3i7kesayatu
*the**conjugacy**classes**of**cyclic**subgroups**of**the**general**linear**group*. As a result, we are able to focus*the*study*of**cyclic**orbit**codes*to a restricted family*of*them. ...*Orbit**codes*are a family*of**codes*employable for communications*on*a random*linear*network*coding*channel.*The*paper focuses*on**the*classification*of*these*codes*. ...*The*main result is that we can focus*on**the*study*of**cyclic**orbit**codes*defined by a*cyclic**group**generated*by a matrix in rational canonical form. ...##
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A subspace code of size \begin{document}$ \bf{333} $\end{document} in the setting of a binary \begin{document}$ \bf{q} $\end{document}-analog of the Fano plane

2019
*
Advances in Mathematics of Communications
*

Moreover, if a

doi:10.3934/amc.2019029
fatcat:nkclljr5vngjtl5b4ww4hbulvm
*code*with these parameters has at least 333 elements, its automorphism*group*is in*one**of*31*conjugacy**classes*. ... This is achieved by a more*general*technique for an exhaustive search in a finite*group*that does not depend*on**the*enumeration*of*all*subgroups*. ... In*the*remainder*of*this paper, we will simply speak*of**the*dimension*of**the*fixed-point space*and*use it in*the*context*of**cyclic**groups**and*their*conjugacy**classes*. ...##
###
Constructing permutation representations for large matrix groups

1994
*
Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94
*

*The*key idea is to build

*the*new permutation representation using

*the*conjugation action

*on*a

*conjugacy*

*class*

*of*

*subgroups*

*of*prime order. ... A unique signature for each

*group*element corresponding to

*the*

*conjugacy*

*class*is used in order to avoid matrix multiplication. ...

*The*authors especially acknowledge that

*the*matrix

*generators*for Lyons's

*group*, for which these techniques were tested, were constructed by Robert Wilson

*and*provided to us by Klaus Lux. ...

##
###
On the automorphism group of a binary q-analog of the Fano plane

2016
*
European journal of combinatorics (Print)
*

Up to

doi:10.1016/j.ejc.2015.07.014
fatcat:mlgwbsrgq5csvosho7m7d3pemq
*conjugacy*in GL(7,2), there remains a single possible*group**of*order 2*and*4, respectively,*and*two possible*groups**of*order 3. ... As a conclusion, it is either rigid or its automorphism*group*is*cyclic**of*order 2, 3 or 4. ...*The*authors are grateful to*the*organizers*of**the*conference "Conference*on*Random Network*Codes**and*Designs over GF(q)" held at*the*Department*of*Mathematics*of*Ghent University, Belgium, from September ...##
###
Update: Some new results on lower bounds on (n,r)-arcs in PG(2,q) for q≤ 31
[article]

2021
*
arXiv
*
pre-print

all would only admit

arXiv:2106.05908v1
fatcat:kcdn2xpnerh5te474jopitknyq
*the*trivial automorphism*group**of*order 1. ... An (n,r)-arc in PG(2,q) is a set B*of*points in PG(2,q) such that each line in PG(2,q) contains at most r elements*of*B*and*such that there is at least*one*line containing exactly r elements*of*B. ... In*the*following, a transversal*of**conjugacy**classes**of**cyclic**subgroups**of*PGL(3, q) will be abbreviated by Conj(q). ...##
###
A design and a geometry for the group Fi 22

2007
*
Designs, Codes and Cryptography
*

This design does not have a doubly transitive automorphism

doi:10.1007/s10623-007-9041-1
fatcat:lm3gvmujtfh75hualzyxknmqci
*group*, since there is a partial*linear*space with lines*of*size 4 defined combinatorially from*the*design*and*preserved by its automorphism*group*... We investigate this geometry*and*determine*the*structure*of*various subspaces*of*it. ... It is*the*normalizer*of**the**cyclic**subgroup**generated*by an element g*of*order 3 in*the**conjugacy**class*3B (in ATLAS [3] notation). ...##
###
Random generation of linear codes

1999
*
Aequationes Mathematicae
*

Dedicated to Professor János Aczél

doi:10.1007/s000100050107
fatcat:fa52uh6pabdmli6exu4wufsh4i
*on**the*occasion*of*his birthday. Abstract Isometry*classes**of**linear**codes*can be expressed as*orbits*under*the**group*action*of*a wreath product. ... Some combinatorial*and*algebraic methods are discussed which can be applied for*generating**linear**codes*distributed uniformly at random over all isometry*classes*. * Supported by*the*Fonds zur Förderung ... Acknowledgment:*The*author wants to thank both Prof. Adalbert Kerber*and*Prof. Jens Schwaiger for their guidance*and*support while preparing this article. ...##
###
Random generation of linear codes

1999
*
Aequationes Mathematicae
*

Dedicated to Professor János Aczél

doi:10.1007/s000100050021
fatcat:typdjmm6lzczne577k3asdrkru
*on**the*occasion*of*his birthday. Abstract Isometry*classes**of**linear**codes*can be expressed as*orbits*under*the**group*action*of*a wreath product. ... Some combinatorial*and*algebraic methods are discussed which can be applied for*generating**linear**codes*distributed uniformly at random over all isometry*classes*. * Supported by*the*Fonds zur Förderung ... Acknowledgment:*The*author wants to thank both Prof. Adalbert Kerber*and*Prof. Jens Schwaiger for their guidance*and*support while preparing this article. ...##
###
Construction of Finite Groups

1999
*
Journal of symbolic computation
*

*The*second algorithm can be used to determine

*the*

*groups*

*of*order p n · q with a normal Sylow

*subgroup*for distinct primes p

*and*q. ... We introduce three practical algorithms to construct certain finite

*groups*up to isomorphism.

*The*first

*one*can be used to construct all soluble

*groups*

*of*a given order. ... Short (1992) has developed an algorithm to compute up to

*conjugacy*all soluble irreducible

*subgroups*

*of*a

*general*

*linear*

*group*. ...

##
###
Niederreiter cryptosystems using quasi-cyclic codes that resist quantum Fourier sampling

2021
*
Advances in Mathematics of Communications
*

We prove, if these quasi-

doi:10.3934/amc.2021062
fatcat:3mppxf52obbt7lenjgszourog4
*cyclic**codes*satisfy certain conditions,*the*corresponding Niederreiter cryptosystem is resistant to*the*hidden*subgroup*problem using weak quantum Fourier sampling. ... <p style='text-indent:20px;'>McEliece*and*Niederreiter cryptosystems are robust*and*versatile cryptosystems. These cryptosystems work with many*linear*error-correcting*codes*. ... We,*the*authors, owe a debt*of*gratitude to both*the*referees for their thorough reading*of**the*manuscript*and*insightful comments. This has improved*the*content*of*this paper*and*its presentation. ...##
###
A note on regular subgroups of the automorphism group of the linear Hadamard code
[article]

2018
*
arXiv
*
pre-print

We consider

arXiv:1807.04446v1
fatcat:d4qz7b5fw5fmxkxjtzzsbhhs6u
*the*regular*subgroups**of**the*automorphism*group**of**the**linear*Hadamard*code*. ... These*subgroups*correspond to*the*regular*subgroups**of*GA(r,2), w.t.r action*on**the*vectors*of*F_2^r, where n=2^r-1 is*the*length*of**the*Hamadard*code*. ... Propelinear*codes*provide a*general*view*on**linear**and*additive*codes*, many*of*which are optimal. ...##
###
Computing the character table of a finite group

2006
*
Journal of symbolic computation
*

We report

doi:10.1016/j.jsc.2006.04.002
fatcat:4yuc72rmivadrov6g6saztjrty
*on**the*performance*of**the*algorithm in computing character tables*of*various*groups*, including local*subgroups**of*3Fi 24*and*maximal*subgroups**of**the*Monster. ... We describe an implementation*of*such an algorithm*and*report*on*its performance.*The*main ingredients*of**the*algorithm are Brauer's theorem,*and*lattice reduction by LLL. ... Acknowledgements I would like to thank John Cannon for his enthusiastic support*of*this project*and*Eamonn O'Brien for supplying*the**groups**of*Fig. 6 . ...##
###
On the Conjugacy Problem in Groups and its Variants
[article]

2010
*
arXiv
*
pre-print

We analyze recent results by Bogopolski, Martino, Maslakova

arXiv:1004.3588v1
fatcat:p7mpq3pvrfbeba4qwi4dkpm764
*and*Ventura*on**the*twisted*conjugacy*problem in free*groups**and*its implication for*the**conjugacy*problem in free-by-*cyclic**groups**and*some further ... This thesis deals with*the**conjugacy*problem in*groups**and*its twisted variants. ... Ventura*on**the*solvability*of**the**conjugacy*problem in free-by-*cyclic**groups*[6] . ...##
###
On Diameter of Subgraphs of Commuting Graph in Symplectic Group for Elements of Order Three

2021
*
Sains Malaysiana
*

Additionally, two mathematical formulas are derived

doi:10.17576/jsm-2021-5002-25
fatcat:mwb4ahoy7ffk5cjmrdfvxn34da
*and*proved,*one*gives*the*number*of*subgraphs based*on**the*size*of*each subgraph*and**the*size*of**the**conjugacy**class*, whilst*the*other*one*gives*the*...*The*aim*of*this paper was to describe*the*structure*of*disconnected commuting graph by considering a symplectic*group**and*a*conjugacy**class**of*elements*of*order three. ... We are also indebted to*the*referees for their suggestions*and*useful remarks. ...##
###
McKay's correspondence and characters of finite subgroups of SU(2)
[article]

2003
*
arXiv
*
pre-print

According to McKay (1980)

arXiv:math/0307121v1
fatcat:tdwchb7w7fchxfsoznh3qv3bxa
*the*irreducible characters*of*finite*subgroups**of*SU(2) are in a natural 1-1 correspondence with*the*extended Coxeter-Dynkin graphs*of*type ADE. ... We show that*the*character values themselves can be given by an uniform formula, as special values*of*polynomials which arise naturally as numerators*of*Poincare series associated to finite*subgroups**of*... A*conjugacy**class**of*Hs corresponds to either 1 or 2*orbits**of*P s, depending*on*whether P*and*P − belong to*the*same*orbit*or not. ...
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