Affinity of permutations of F2n

Authors:
Xiang-dong Hou
Affiliations:
Department of Mathematics, University of South Florida, Tampa, FL and Department of Mathematics and Statistics, Wright State University, Dayton, OH
Venue:
Discrete Applied Mathematics - Special issue: Coding and cryptography
Year:
2006

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Abstract

It was conjectured that if n is even, then every permutation of F2n is affine on some 2-dimensional affine subspace of F2n. We prove that the conjecture is true for n = 4, for quadratic permutations of F2n and for permutation polynomials of F2n with coefficients in F2n/2. The conjecture is actually a claim about (AGL(n, 2), AGL(n, 2))-double cosets in permutation group S(F2n) of F2n. We give a formula for the number of (AGL(n, 2), AGL(n, 2))-double cosets in S(F2n) and classify the (AGL(4, 2), AGL(4, 2))-double cosets in S(F24).