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

Felice Manganiello, Anna-Lena Trautmann, Joachim Rosenthal
2011 2011 IEEE International Symposium on Information Theory Proceedings  
We start by classifying 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.  ... 
doi:10.1109/isit.2011.6033885 dblp:conf/isit/ManganielloTR11 fatcat:q67oir42yngonfi3i7kesayatu

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

Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann
2019 Advances in Mathematics of Communications  
Moreover, if a 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.  ... 
doi:10.3934/amc.2019029 fatcat:nkclljr5vngjtl5b4ww4hbulvm

Constructing permutation representations for large matrix groups

Gene Cooperman, Larry Finkelstein, Bryant York, Michael Tselman
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.  ... 
doi:10.1145/190347.190384 dblp:conf/issac/CoopermanFYT94 fatcat:f25nr465rzgcpcbac7xygt25yq

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

Michael Braun, Michael Kiermaier, Anamari Nakić
2016 European journal of combinatorics (Print)  
Up to 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  ... 
doi:10.1016/j.ejc.2015.07.014 fatcat:mlgwbsrgq5csvosho7m7d3pemq

Update: Some new results on lower bounds on (n,r)-arcs in PG(2,q) for q≤ 31 [article]

Michael Braun
2021 arXiv   pre-print
all would only admit 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).  ... 
arXiv:2106.05908v1 fatcat:kcdn2xpnerh5te474jopitknyq

A design and a geometry for the group Fi 22

P. J. Cameron, A. Rudvalis
2007 Designs, Codes and Cryptography  
This design does not have a doubly transitive automorphism 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).  ... 
doi:10.1007/s10623-007-9041-1 fatcat:lm3gvmujtfh75hualzyxknmqci

Random generation of linear codes

H. Fripertinger
1999 Aequationes Mathematicae  
Dedicated to Professor János Aczél 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.  ... 
doi:10.1007/s000100050107 fatcat:fa52uh6pabdmli6exu4wufsh4i

Random generation of linear codes

H. Fripertinger
1999 Aequationes Mathematicae  
Dedicated to Professor János Aczél 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.  ... 
doi:10.1007/s000100050021 fatcat:typdjmm6lzczne577k3asdrkru

Construction of Finite Groups

H.U. Besche, B. Eick
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.  ... 
doi:10.1006/jsco.1998.0258 fatcat:limqcxsqanfrplrnd6rzgo3k44

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

Upendra Kapshikar, Ayan Mahalanobis
2021 Advances in Mathematics of Communications  
We prove, if these quasi-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.  ... 
doi:10.3934/amc.2021062 fatcat:3mppxf52obbt7lenjgszourog4

A note on regular subgroups of the automorphism group of the linear Hadamard code [article]

I.Yu.Mogilnykh
2018 arXiv   pre-print
We consider 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.  ... 
arXiv:1807.04446v1 fatcat:d4qz7b5fw5fmxkxjtzzsbhhs6u

Computing the character table of a finite group

W.R. Unger
2006 Journal of symbolic computation  
We report 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 .  ... 
doi:10.1016/j.jsc.2006.04.002 fatcat:4yuc72rmivadrov6g6saztjrty

On the Conjugacy Problem in Groups and its Variants [article]

Michèle Feltz
2010 arXiv   pre-print
We analyze recent results by Bogopolski, Martino, Maslakova 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] .  ... 
arXiv:1004.3588v1 fatcat:p7mpq3pvrfbeba4qwi4dkpm764

On Diameter of Subgraphs of Commuting Graph in Symplectic Group for Elements of Order Three

Suzila Mohd Kasim, Athirah Nawawi
2021 Sains Malaysiana  
Additionally, two mathematical formulas are derived 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.  ... 
doi:10.17576/jsm-2021-5002-25 fatcat:mwb4ahoy7ffk5cjmrdfvxn34da

McKay's correspondence and characters of finite subgroups of SU(2) [article]

Wulf Rossmann
2003 arXiv   pre-print
According to McKay (1980) 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.  ... 
arXiv:math/0307121v1 fatcat:tdwchb7w7fchxfsoznh3qv3bxa
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