Results on permutations with distinct difference property

We prove that for all odd primes $p$ and positive integers $alpha geq 2$, a construction of Batten and Sane yields at least $(p-1)^3/4$ permutations with a distinct difference property (DDP) of ${1,2,ldots,p^alpha-1}$. This proves a conjecture of Batten and Sane, that at least $(p-1)^2/2$ such permutations exist. We also pose several research questions for DDP permutations.