Abstract
Dillon and Dobbertin proved that if L := GF(2m), gcd(k, m) = 1, d := 4k − 2k + 1 and Δ k (x) := (x + 1)d + x d + 1, then B k := L\Δ k (L) is a difference set in the cyclic multiplicative group L × of L. Used in the proof were the auxiliary functions \(c_k^{\gamma}(x) := b_k(\gamma x^{2^k+1})\) , where γ is in L × and b k is the characteristic function of B k on L. When m is odd \(c_k^{\gamma}\) is itself the characteristic function of a cyclic difference set which is equivalent to B k . In this paper we point out that when m is even and γ is not a cube in L then \(c_k^{\gamma}\) is the characteristic function of a difference set in the elementary abelian additive group of L; i.e. \(c_k^{\gamma}\) is a bent function.
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In memory of our friend and colleague Hans Dobbertin.
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Dillon, J.F. More DD difference sets. Des. Codes Cryptogr. 49, 23–32 (2008). https://doi.org/10.1007/s10623-008-9188-4
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DOI: https://doi.org/10.1007/s10623-008-9188-4