Difference sets in Z(4)(m) and F-2(2m)
Let R = GR(4, m) be the Galois ring of cardinality 4(m) and let T be the Teichmuller system of R. For every map lambda of T into {-1,+1} and for every permutation pi of T, we define a map phi(lambda)(pi) of R into {-1,+1} as follows: if x is an element of R and if x = a + 2b is the 2-adic representation of x with a is an element of T and b is an element of T, then phi(lambda)(pi)(x) = lambda(a) + 2Tr(pi(a)b), where Tr is the trace function of R . For i = 1 or i = -1, define D-i as the set of x in R such that phi(lambda)(pi) = i. We prove the following results: 1) D-i is a Hadamard difference set of (R, +). 2) If Phi is the Gray map of R into F-2(2m), then Phi(D-i) is a difference set of (F-2(2m), +). 3) The set of D-i and the set of Phi(D-i) obtained for all maps lambda and pi, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D-i in the additive group of R. Examples are given by means of morphisms and norm in R.
difference sets galois ring gray map