On the minimum size of signed sumsets in elementary abelian groups

For a finite abelian group G and positive integers m and h, we letρ(G,m,h)=min⁡{|hA|:A⊆G,|A|=m} andρ±(G,m,h)=min⁡{|h±A|:A⊆G,|A|=m}, where hA   and h±Ah±A denote the h-fold sumset and the h-fold signed sumset of A  , respectively. The study of ρ(G,m,h)ρ(G,m,h) has a 200-year-old history and is now known for all G, m, and h  . In previous work we provided an upper bound for ρ±(G,m,h)ρ±(G,m,h) that we believe is exact, and proved that ρ±(G,m,h)ρ±(G,m,h) agrees with ρ(G,m,h)ρ(G,m,h) when G   is cyclic. Here we study ρ±(G,m,h)ρ±(G,m,h) for elementary abelian groups G; in particular, we determine all values of m   for which ρ±(Zp2,m,2) equals ρ(Zp2,m,2) for a given prime p.