Abstract
Using the classification of finite simple groups, we obtain a condition for a permutation group on a finite field GF(pm) to have the affine group AGL(1,pm) as a subgroup. Applying this result to the automorphism groups of non trivial affine-invariant codes, we prove that these automorphism groups are always subgroups of the general affine group AGL(m,p).
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References
T Berger, A direct proof for the Automorphism Group of Reed Solomon Codes, Proceedings of Eurocode'90 (G Cohen and P Charpin ed.), Lecture notes in Computer science, Springer Verlag 514, Berlin-Hieidelberg, New York (1991) pp 21–29
W Bumside, Theory of Groups of Finite Order Cambridge University Press, Cambridge (1911).
PJ Cameron, Finite Permutation Groups ld Finite Simple Groups, Bull London Math Soc. n 13 (1981) p 1–22.
P Charpin, Codes Cycliques Etendus Invariants sous le Groupe Affine, These de Doctorat d'Etat, Universite Paris VII. LITP (1987)
P Charpin, The Extended Reed Solomon Codes considered as Ideals of a Modular Algebra, Annals of Discrete Mathematics n.17 (1983), p 171–176.
J. Dieudonna, On the Automorphisms of the Classical Groups, Memoirs of the American Mathematical Society n.2 (1951).
A. Dur, The Automorphism Group of Reed Solomon Codes, Journal of Combinatorial Theory series A, vol. 4, 1 (1987).
R.M Guralnick, Subgroups of Prime Power index in a simple group, Journal of Algebran.81 (1983), p. 304–311.
B. Huppert, Endliche Gruppen 1 Springer Verlag 134, Berlin-Heidelberg, New York (1967).
B. Huppert and N Blackburn, Finite Groups IISpringer-Verlag 242, Berlin-Heidelberg, New York (1982)
B. Huppert and N. Blackburn, Finite Groups 111 Springer-Verlag 243, Berlin-Heidelberg, New York (1982).
T.Kasami, S.Lin and W.W. Peterson, Some Results on Cyclic Codes which are Invariant under the Affine Gloup and their Applications, Info and Control, vol.11 (1967), pp475–496
W. Knapp and P Schmidt, Codes with Prescribed Permutation Group, Journal of Algebra n67 (1980) pp.415–435
F.J MacWilliams and NJA Sloane, The Theory of Error Correcting Codes North Holland, Amsterdam (1977)
B Mortimer, Permutation Groups Containing Affine Groups of the same Degree Journal of London Math Soc vol 2. n 15 (1977) pp.445–455.
M. Suzuki, Group Theory I Springer-Verlag 247, Berlin Heidelberg, New York(1982)
H Wielandt, Finite Permutation Groups, Academic Press, New York and London (1964)
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Berger, T.P. On the Automorphism Groups of Affine-Invariant Codes.. Designs, Codes and Cryptography 7, 215–221 (1996). https://doi.org/10.1023/A:1018038723045
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DOI: https://doi.org/10.1023/A:1018038723045