  The binary $k$-dimensional simplex code is known to be a $2^{k-1}$-batch code
and is conjectured to be a $2^{k-1}$-functional batch code. Here, we offer a
simple, constructive proof of a result that is "in between" these two
properties. Our approach is to relate these properties to certain (old and new)
additive problems in finite abelian groups. We also formulate a conjecture for
finite abelian groups that generalizes the above-mentioned conjecture.
