Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2025-01-05T12:29:03.707Z Has data issue: false hasContentIssue false

Some Remarks on Self-Dual Codes Invariant Under Almost Simple Permutation Groups

Published online by Cambridge University Press:  15 April 2019

By
B. G. Rodrigues  and
T. M. Mudziiri Shumba
Edited by
C. M. Campbell ,
C. W. Parker ,
M. R. Quick ,
E. F. Robertson  and
C. M. Roney-Dougal
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
Get access

Summary

We present some recent advances on the question of enumeration and classification of self-dual codes invariant under finite non-solvable permutation groups. In particular, we examine the question for almost simple groups of sporadic or rank 3 types. As a by-product we determine all doubly even and extremal codes of length n ≤ 4095 which admit an almost simple group of sporadic or rank 3 type as a permutation group of automorphisms acting transitively on the code’s coordinate positions.

Type
Chapter
Information
Groups St Andrews 2017 in Birmingham , pp. 469 - 485
Publisher: Cambridge University Press
Print publication year: 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bosma, W., Cannon, J., Playoust, C.. The Magma algebra system I: The user language. J. Symbolic Comput., 24 (1997), 235–265.CrossRefGoogle Scholar
Bosma, W. and Cannon, J.. Handbook of Magma Functions. Department of Mathematics, University of Sydney, 1994. http://www.maths.usyd.edu.au:8000/u/magma/.Google Scholar
Breuer, Thomas. Decomposition Matrices. Available at: The Modular Atlas homepage. http://www.math.rwth-aachen.de/∼MOC/decomposition 1999.Google Scholar
Chigira, N., Harada, M. and Kitazume, M.. Permutation groups and binary selforthogonal codes. J. Algebra, 309 (2007), 610–621.CrossRefGoogle Scholar
Chigira, N., Harada, M. and Kitazume, M.. Some self-dual codes invariant under the Hall-Janko group. J. Algebra, 316 (2007), 578–590.CrossRefGoogle Scholar
Chigira, N., Harada, M. and Kitazume, M.. On the classification of extremal doubly even self-dual codes with 2-transitive automorphism groups. Des. Codes Cryptogr., 73 (2014), 33–35.CrossRefGoogle Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., and Wilson, R. A.. An Atlas of Finite Groups. Oxford: Oxford University Press, 1985.Google Scholar
Connor, T. and Leemans, D.. An atlas of subgroup lattices of finite almost simple groups. Ars Math. Contemp., 8 (2015), 259–266.CrossRefGoogle Scholar
Connor, T. and Leemans, D.. The Subgroup Lattice of M11. http://homepages.ulb.ac.be/∼dleemans/atlaslat/m11.pdf.Google Scholar
Devillers, A., Giudici, M., Li, C. H., Pearce, G. and Praeger, C. E.. On imprimitive rank 3 permutation groups. J. London Math. Soc., 84 (2011), 649-–669.CrossRefGoogle Scholar
Devillers, A., Giudici, M., Li, C. H., Pearce, G. and Praeger, C. E.. Correction to “On imprimitive rank 3 permutation groups,” J. London Math. Soc. 84 (2011), 649–669.Google ScholarGoogle Scholar
GAP. Groups, Algorithms and Programming, Version 4. The GAP Group, Lehrstuhl D für Mathematik, RWTH Aachen, Germany and School of Computer Sciences, University of St Andrews, Scotland. http://www.gap-system.org/.Google Scholar
Gleason, A.M.. Weight polynomials of self-dual codes and the MacWilliams identities. In: Congrés, Actes du International des Mathématiciens (Nice, 1970), Tome 3, pp. 211–215. Gauthier-Villars, Paris (1971).Google Scholar
Günther, A. and Nebe, G.. Automorphisms of doubly even self-dual codes. Bull. London Math. Soc., 41 (2009), 769–778.CrossRefGoogle Scholar
Cary Huffman, W.. On the classification and enumeration of self-dual codes. Finite Fields and Their Applications, 11 (2005), 451–490.CrossRefGoogle Scholar
Huppert, B.. Endliche Gruppen I. Berlin, Heidelberg: Springer Verlag, 1967.CrossRefGoogle Scholar
Lataille, J. M., Peter Sin and Pham Huu Tiep. The modulo 2 structure of rank 3 permutation modules for odd characteristic symplectic groups. J. Algebra, 268 (2003), 463–483.CrossRefGoogle Scholar
Malevich, A.. Extremal self-dual codes. Ph.D. Thesis, Otto-von-Guericke-Universität Magdeburg, 2012.Google Scholar
Malevich, A. and Willems, W.. On the classification of the extremal self-dual codes over small fields with 2-transitive automorphism groups. Des. Codes Cryptogr. 70 (2014), 69–76.CrossRefGoogle Scholar
Mallows, C. L. and Sloane, N.J.A.. An upper bound for self-dual codes. Inf. Control, 22 (1973), 188–200.CrossRefGoogle Scholar
Moori, J. and Rodrigues, B. G.. Some designs and binary codes preserved by the simple group Ru of Rudvalis. J. Algebra, 372 (2012), 702–710.CrossRefGoogle Scholar
Mukwembi, S., Rodrigues, B. G. and Shumba, T. M.. On self-dual binary codes invariant under permutation groups. Submitted.Google Scholar
Parker, R. A. and Wilson, R. A.. The 2-modular character table of the Rudvalis group, unpublished, 1998.Google Scholar
Rains, E. M.. Shadow bounds for self-dual-codes. IEEE Trans. Inf. Theory, 44 (1998), 134–139.CrossRefGoogle Scholar
Rains, E.M. and Sloane, N.J.A.. Self-dual codes. In Pless, V. S. and Huffman, W. C., editors, Handbook of Coding Theory, pages 177–294. Amsterdam: Elsevier, 1998. Volume I and II.Google Scholar
Rodrigues, B. G.. A classification of binary self-dual codes with a primitive rank 3 automorphism group of almost simple type. Submitted.Google Scholar
Rodrigues, B. G.. On self-dual codes invariant under 2-transitive groups. In preparation.Google Scholar
Rodrigues, B. G.. A classification of binary self-dual codes admitting an imprimitive rank 3 automorphism group of almost simple type. Submitted.Google Scholar
Zhang, S.. On the nonexistence of extremal self-dual codes. Discrete Appl. Math., 91 (1999), 277–286.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×