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A POLYNOMIAL APPROACH TO COCYCLES OVER ELEMENTARY ABELIAN GROUPS

Part of: Connections with homological algebra and category theory Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  01 October 2008

D. G. FARMER  and
K. J. HORADAM
D. G. FARMER
Affiliation:
RMIT University, Melbourne, VIC 3001, Australia (email: [email protected])
K. J. HORADAM*
Affiliation:
RMIT University, Melbourne, VIC 3001, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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We derive bivariate polynomial formulae for cocycles and coboundaries in Z2(ℤpn,ℤpn), and a basis for the (pn−1−n)-dimensional GF(pn)-space of coboundaries. When p=2 we determine a basis for the -dimensional GF(2n)-space of cocycles and show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form.

Keywords

2-cocyclecoboundarymultiplicative cocyclebasisPascal’s triangle

MSC classification

Secondary: 11T06: Polynomials 20J06: Cohomology of groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 177 - 190
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Batten, Lynn M., Coulter, Robert S. and Henderson, Marie, ‘Extending abelian groups to rings’, J. Aust. Math. Soc. 82 (2007), 297–313.Google Scholar
[2]Brown, Kenneth S., Cohomology of Groups, Geometry and Topology Monographs, 87 (Springer, New York, 1982).Google Scholar
[3]Budaghyan, Lilya, Carlet, Claude and Pott, Alexander, ‘New classes of almost bent and almost perfect nonlinear polynomials’, IEEE Trans. Inform. Theory 52 (2006), 1141–1152.Google Scholar
[4]Canteaut, Anne, ‘Cryptographic functions and design criteria for block ciphers’, in: INDOCRYPT 2001, Lecture Notes in Computer Science, 2247 (eds. C. Pandu Rangan and C. Ding) (Springer, Berlin, 2001), pp. 1–16.Google Scholar
[5]Coulter, Robert S. and Matthews, Rex W., ‘Planar functions and planes of Lenz–Barlotti Class II’, Des. Codes Cryptogr. 10 (1997), 167–184.Google Scholar
[6]Fine, N. J., ‘Binomial coefficients modulo a prime’, Amer. Math. Monthly 54 (1947), 589–592.Google Scholar
[7]Flannery, D. L. and O’Brien, E. A., ‘Computing 2-cocycles for central extensions and relative difference sets’, Comm. Algebra 28 (2000), 1939–1955.CrossRefGoogle Scholar
[8]Horadam, K. J., Hadamard Matrices and Their Applications (Princeton University Press, Princeton, NJ, 2006).Google Scholar
[9]Horadam, K. J. and Farmer, D. G., ‘Bundles, presemifields and nonlinear functions’, Des. Codes Cryptogr. 48 (2008), 79–94.CrossRefGoogle Scholar
[10]Horadam, Kathy J. and Udaya, Parampalli, ‘Cocyclic Hadamard codes’, IEEE Trans. Inform. Theory 46 (2000), 1545–1550.Google Scholar
[11]Horadam, Kathy J. and Udaya, Parampalli, ‘A new construction of central relative (p a,p a,p a,1)-difference sets’, Des. Codes Cryptogr. 27 (2002), 281–295.Google Scholar
[12]Horadam, Kathy J. and Udaya, Parampalli, ‘A new class of ternary cocyclic Hadamard codes’, Appl. Algebra Engrg. Comm. Comput. 14 (2003), 65–73.Google Scholar
[13]Kyureghyan, Gohar M., ‘Crooked maps in 𝔽2n’, Finite Field Appl. 13 (2007), 713–726.Google Scholar
[14]LeBel, Alain, ‘Shift actions on 2-cocycles’, PhD Thesis, RMIT University, Melbourne, Australia, 2005.Google Scholar
[15]Lidl, Rudolf and Niederreiter, Harald, Finite Fields, 2nd edn (Cambridge University Press, Cambridge, 1997).Google Scholar
[16]Lucas, Édouard, Théorie des Nombres, Tome premier (Gauthier-Villars, Paris, 1891), pp. 417–420 (Reprinted by Albert Blanchard, 1961).Google Scholar
[17]Wolfram, Stephen, ‘Geometry of binomial coefficients’, Amer. Math. Monthly 91 (1984), 566–571.Google Scholar