I have a problem linking the multiplication in a finite field with its additive structure:
The set $S$ is an additive subgroup of $\mathbb{F}_{2^h}$ (the finite field of order $2^h$). I have two different cosets, $\beta+S$ and $\gamma+S$. The question is, whether there is there always an element $\delta$, different from $0,1$ such that $$\delta(\beta+S)\cap (\gamma+S)=\emptyset.$$ In other words, can I multiply the coset with some element $\delta\neq 0,1$ and still be disjoint from the other coset?
In the case I was interested in (a particular additive subgroup of index $4$), I checked by computer for $h=4,5,6$ and there, it is always possible to find such a $\delta$. But I wonder whether there is a general reason for this, independent of the additive subgroup itself and the primitive polynomial.
Edit: the subgroup S I mentioned is simply the additive subgroup generated by $1,\alpha,\alpha^2$, where $\alpha$ is the primitive element $Z(2^h)$ that GAP uses. So I took $\beta=\alpha^3$ and $\gamma=\alpha^4$.