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Code Equivalence and Group Isomorphism [chapter]

László Babai, Paolo Codenotti, Joshua A. Grochow, Youming Qiao
<span title="2011-01-23">2011</span> <i title="Society for Industrial and Applied Mathematics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ja7dpjgz2nhjlovlil3ij5m7ei" style="color: black;">Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms</a> </i> &nbsp;
One of our tools is an algorithm for equivalence of (not necessarily linear) codes in simply-exponential time in the length of the code, obtained by modifying Luks's algorithm for hypergraph isomorphism  ...  We relate the results to the filtration of groups introduced by Babai and Beals (1999).  ...  We reduce the isomorphism problem for semisimple groups to equivalence of group codes. We consider the code equivalence problem as a separate problem of interest in its own right.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/1.9781611973082.107">doi:10.1137/1.9781611973082.107</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/soda/BabaiCGQ11.html">dblp:conf/soda/BabaiCGQ11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/md5cdhszqbem5axmnfsmohnf6a">fatcat:md5cdhszqbem5axmnfsmohnf6a</a> </span>
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Codes and arrays from cocycles

Garry Hughes
<span title="">2001</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qea6koy5crhvfgkjwgnwojjkrm" style="color: black;">Bulletin of the Australian Mathematical Society</a> </i> &nbsp;
In the theory of error correcting codes, cohomology allows us to give an isomorphism between a group ring code and the direct sum of other group ring codes that have been "twisted" by cocycles.  ...  We use the isomorphism to construct new codes from known "smaller" codes.  ...  This idea is used to prove the non-existence of self-dual group ring codes in the case when the group is the direct product of a 2-group and a group of odd order, and the ring is either GF(q), with q odd  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0004972700019420">doi:10.1017/s0004972700019420</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gzakua6325ahpknqjisgxyrsi4">fatcat:gzakua6325ahpknqjisgxyrsi4</a> </span>
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On the number of non-G-equivalent minimal abelian codes [article]

Fatma Altunbulak Aksu, İpek Tuvay
<span title="2020-06-25">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Ferraz, Guerreiro and Polcino Milies prove that the number of G-equivalence classes of minimal abelian codes is equal to the number of G-isomorphism classes of subgroups for which corresponding quotients  ...  As an application, we calculate the number of non-G-equivalent minimal abelian codes for some specific family of abelian groups.  ...  Moreoever as defined in [4] , two abelian codes I and J are called G-equivalent if there is a group automorphism ϕ : G → G whose linear extension to the group algebra maps I onto J.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1904.04077v3">arXiv:1904.04077v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3yzuzohduvhu5ltgkpyb2snj7u">fatcat:3yzuzohduvhu5ltgkpyb2snj7u</a> </span>
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Polynomial-Time Isomorphism Test for Groups with No Abelian Normal Subgroups [chapter]

László Babai, Paolo Codenotti, Youming Qiao
<span title="">2012</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
twisted code equivalence problem," a generalization of the classical code equivalence problem by admitting a group action on the alphabet.  ...  Two key new ingredient are: (a) an algorithm to test permutational isomorphism of permutation groups in time, polynomial in the order and simply exponential in the degree; (b) the introduction of the "  ...  groups, via "twisted code equivalence."  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-31594-7_5">doi:10.1007/978-3-642-31594-7_5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qkmh5ql4mraoxaaird72svte54">fatcat:qkmh5ql4mraoxaaird72svte54</a> </span>
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On the number of non-G-equivalent minimal abelian codes

<span title="">2020</span> <i title="The Scientific and Technological Research Council of Turkey"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3i63uqcqmvbdrbc6b26krcncii" style="color: black;">Turkish Journal of Mathematics</a> </i> &nbsp;
Ferraz, Guerreiro and Polcino Milies prove that the number of G -equivalence 4 classes of minimal abelian codes is equal to the number of G -isomorphism classes of subgroups for which corresponding 5 quotients  ...  As an application, we calculate the number 7 of non-G -equivalent minimal abelian codes for some specific family of abelian groups.  ...  Theorem 1.7 gives a complete characterization of the groups having 21 τ (G) non-G-equivalent minimal abelian codes, that is having the least possible number of non-G-equivalent 22 abelian codes.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3906/mat-2006-102">doi:10.3906/mat-2006-102</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dj7arxyx6bcbbnsg27is6eot6q">fatcat:dj7arxyx6bcbbnsg27is6eot6q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20201212075416/https://online-journals.tubitak.gov.tr/openAcceptedDocument.htm?fileID=1397933&amp;no=301175" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ea/a0/eaa0b68991b8b736799582cf1d47a969e42e49c1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3906/mat-2006-102"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

On Isomorphism Testing of Groups with Normal Hall Subgroups

You-Ming Qiao, Jayalal Sarma M.N., Bang-Sheng Tang
<span title="">2012</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5acgllsvrbf5dar4dbo3vyrz2a" style="color: black;">Journal of Computer Science and Technology</a> </i> &nbsp;
code isomorphism problem.  ...  determine if there exists an automorphism φ : H → H, such that the induced representation ρ φ = ρ • φ and τ are equivalent, in time poly(|H|, p d ).  ...  Acknowledgement Part of the work was done while Youming Qiao was visiting the University of Chicago, and he would like to thank Laci Babai and Sasha Razborov for hosting him.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11390-012-1255-7">doi:10.1007/s11390-012-1255-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nh7bh56dpbar7gpcsbee66wufa">fatcat:nh7bh56dpbar7gpcsbee66wufa</a> </span>
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A characterization of abelian group codes in terms of their parameters [article]

Fatma Altunbulak Aksu, İpek Tuvay
<span title="2022-01-17">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In 1979, Miller proved that for a group G of odd order, two minimal group codes in 𝔽_2G are G-equivalent if and only they have identical weight distribution.  ...  In this paper, we give a characterization of finite abelian groups so that over a specific set of group codes, equality of important parameters of two codes implies the G-equivalence of these two codes  ...  Then two codes I H and I K in I FqG are G-equivalent if and only if H and K are G-isomorphic. Proof. If H and K are G-isomorphic, then by definition there exists θ ∈ Aut(G) such that θ(H) = K.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.06366v1">arXiv:2201.06366v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/az2d6wuryjhs3nogcizwlfcckm">fatcat:az2d6wuryjhs3nogcizwlfcckm</a> </span>
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Matrix Isomorphism of Matrix Lie Algebras

Joshua A. Grochow
<span title="">2012</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fvcoh2vdcnbppmicpce3jet6r4" style="color: black;">2012 IEEE 27th Conference on Computational Complexity</a> </i> &nbsp;
We show that certain cases of MATISOLIE-for the wide and widely studied classes of semisimple and abelian Lie algebras-are equivalent to GRAPH ISOMORPHISM and LINEAR CODE EQUIVALENCE, respectively.  ...  Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley-Sohoni Geometric Complexity Theory program.  ...  These automorphisms are what cause all the computational difficulties, and allow the equivalences with GRAPH ISOMORPHISM and CODE EQUIVALENCE.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/ccc.2012.34">doi:10.1109/ccc.2012.34</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/coco/Grochow12.html">dblp:conf/coco/Grochow12</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xhu5357knfgzloh6kqfetfrbdy">fatcat:xhu5357knfgzloh6kqfetfrbdy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170811185812/http://frontiersinai.com/turingfiles/June/grochow.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/48/84/4884b97f07dcf37d8cde95954bc2b2a3f11cb5c2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/ccc.2012.34"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Permutation Code Equivalence is not Harder than Graph Isomorphism when Hulls are Trivial [article]

Magali Bardet and Ayoub Otmani and Mohamed Saeed-Taha
<span title="2019-04-30">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that given access to a subroutine that decides if two weighted undirected graphs are isomorphic, one may deterministically decide the permutation code equivalence provided that the underlying vector  ...  This problem is referred to as the permutation code equivalence.  ...  This group is called the automorphism group. The set of isomorphisms between two equivalent linear subspaces is then a coset of the automorphism group.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1905.00073v1">arXiv:1905.00073v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tgaxvjdqsbempmvodgiqip6zk4">fatcat:tgaxvjdqsbempmvodgiqip6zk4</a> </span>
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A solution of an equivalence problem for semisimple cyclic codes [article]

M.Muzychuk
<span title="2011-05-22">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we propose an efficient solution of an equivalence problem for semisimple cyclic codes.  ...  When the set S e is produced an equivalence testing becomes rather simple: a cyclic code J F q [H] is equivalent to the code I = eF q [H] if and only if it is equivalent by a permutation from S e .  ...  This gives a simple algorithm for a code equivalence testing which is polynomial in n and q.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1105.4320v1">arXiv:1105.4320v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ujho2d6chbex7h4opvs4ujs5ra">fatcat:ujho2d6chbex7h4opvs4ujs5ra</a> </span>
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Characters and the equivalence of codes

Harold N. Ward, Jay A. Wood
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
MacWilliams proved that two linear codes are equivalent up to monomial transformations if and only if there is a weight-preserving linear isomorphism between them.  ...  We present a new proof based on the linear independence of group characters. We also present a new proof of a theorem of Bonisoli characterizing constant weight codes.  ...  Two linear codes C~, C 2 of dimension k in K" are equivalent if and only if there exists an abstract K-linear isomorphism f: C1 ---" C2 which preserves weights, wt(f(x)) = wt(x), for all x ~ Cl.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0097-3165(96)80011-2">doi:10.1016/s0097-3165(96)80011-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zbyjxfv24ncxdeu2kwyrllnlcq">fatcat:zbyjxfv24ncxdeu2kwyrllnlcq</a> </span>
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About Code Equivalence – a Geometric Approach [article]

Iliya Bouyukliev, Stefka Bouyuklieva
<span title="2022-02-04">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we present algorithms for equivalence of linear codes, based on their relation to multisets of points in a projective geometry.  ...  The equivalence test is a main part in any classification problem. It helps to prove bounds for the main parameters of the considered combinatorial structures and to study their properties.  ...  We use smaller binary matrices and prove that if these matrices are not isomorphic then the codes are not equivalent.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2202.02086v1">arXiv:2202.02086v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jgyeywc56bbgphuivjpbdo5zdi">fatcat:jgyeywc56bbgphuivjpbdo5zdi</a> </span>
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Automorphism Properties and Classification of Adinkras

B. L. Douglas, S. James Gates, B. L. Segler, J. B. Wang
<span title="">2015</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/dur7nzaqy5hezkanc3eeianfuy" style="color: black;">Advances in Mathematical Physics</a> </i> &nbsp;
Using this, we classify Adinkras according to their equivalence and isomorphism classes.  ...  We extend previous results dealing with characterization of Adinkra degeneracy via matrix products and present algorithms for calculating the automorphism groups of Adinkras and partitioning Adinkras into  ...  Acknowledgment The authors would like to thank Akihiro Munemasa for detailed comments and suggestions on this paper, as well as Cheryl Praeger, Cai-Heng Li, Gordon Royle, and John Bamberg for several useful  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2015/584542">doi:10.1155/2015/584542</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5o7avx262necnaqxsuqvvk6i64">fatcat:5o7avx262necnaqxsuqvvk6i64</a> </span>
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Gödel Diffeomorphisms

Matthew Foreman
<span title="2020-09-01">2020</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nxnim3i24fhcvkvgruk2dej7yu" style="color: black;">Bulletin of Symbolic Logic</a> </i> &nbsp;
As a corollary there is a diffeomorphism that is isomorphic to its inverse if and only if the Riemann Hypothesis holds, a different one that is isomorphic to its inverse if and only if Goldbach's conjecture  ...  holds and so forth.  ...  The Polish group of Measure Preserving Transformations, M P T , acts on itself by conjugacy and the associated equivalence relation is the same as measure isomorphism. Definition 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/bsl.2020.36">doi:10.1017/bsl.2020.36</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kl4fein2sjb7jmvzb4nd2bwlym">fatcat:kl4fein2sjb7jmvzb4nd2bwlym</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200912042457/https://www.cambridge.org/core/services/aop-cambridge-core/content/view/DC6656BB17EDDB657A11C046C7DE18A8/S1079898620000360a.pdf/div-class-title-godel-diffeomorphisms-div.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/bc/1a/bc1ad1168e48c3d46fa87632db78963a0b682f09.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/bsl.2020.36"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> cambridge.org </button> </a>

Lie algebra conjugacy [article]

Joshua A. Grochow
<span title="2011-12-09">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Specifically, we show: Abelian Lie algebra conjugacy is equivalent to the code equivalence problem, and hence is as hard as graph isomorphism.  ...  Affine equivalence is related to many complexity problems such as factoring integers, graph isomorphism, matrix multiplication, and permanent versus determinant.  ...  For codes over F 2 , code equivalence was known to be as hard as graph isomorphism [PR97] ; we extend their proof to show that code equivalence over any field is as hard as graph isomorphism.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1112.2012v1">arXiv:1112.2012v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t2bgagusofcfrp3dqjefzlhlru">fatcat:t2bgagusofcfrp3dqjefzlhlru</a> </span>
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