Difference sets in \(\mathbb{Z}_4^m\) and \(\mathbb{F}_2^{2m}\)
The author gives a construcion of difference sets of the additive group \((R,+)\) of the Galois ring \(R\) with \(4^m\) elements and studies their images under the Gray mapping \(\Phi\) in the vector space \(F_2^{2m}\) over the binary field. The basic construction is as follows: For every mapping \(\lambda\) of the Teichmüller system of \(R\) into the units \(U=\{ -1, 1\}\) of \(Z_4\) and for any permutation \(\pi\) of \(T\) the mapping \(\phi_\lambda^\pi\) from \(R\) into \(U\) is defined by \(\phi_\lambda^\pi(a+2b):=\lambda(a) + 2\)Tr\((\pi(a)b)\), where Tr is the trace mapping of \(R\). For \(i \in U\) the preimage \(D_i=D_{i,\lambda,\pi}\) of \(i\) under \(\phi_\lambda^\pi\) is a Hadamard difference set of \((R,+)\). It is further proved that any \(\Phi(D_i)\) is a Hadamard difference set of \((F_2^{2m},+)\). The author thereby provides a one-to-one correspondence of the sets \(D_{i,\lambda,\pi}\) to the class of binary Maiorana-McFarland difference sets in \((F_2^{2m},+)\). A further interesting result within the ring theoretical setting is that certain multiplicative subgroups of (the units of) \(R\) are difference sets of the form \(D_{i,\lambda,\pi}\).
scientific article; zbMATH DE number 1441593