TITLE: The maximal subset of a finite field where the sum of any subset is non-zero
QUESTION [10 upvotes]: Given a finite field $\mathbb{F}_q$ with $q=p^m$ where $p$ is the characteristic.
For any subset $S=\{a_1,\dots,a_n\}$ of $\mathbb{F}_q$, if any partial sum (i.e. the sum of elements in a non-empty subset of $S$) is non-zero, then we may call $S$ a good subset.
My question is what's the maximal cardinality $f(q)$ of a good subset $S$?
Or are there any (lower) bounds for $f(q)$?

REPLY [9 votes]: To complete Seva's answer, the article (“Subset sums modulo a prime”) of Nguyen, Szemerédi and Vu gives an asymptotic result for $p$ large enough, and this was also independently proved by Deshouillers and Prakash in “Large zero-free subsets of $\mathbb{Z}/p\mathbb{Z}$”, Integers, Procedings
of Integers conference 2009 11A (2011), article 8.
The result in $\mathbb{F}_p$ for any prime $p$ is established in “An addition theorem and maximal zero-sum free sets in $\mathbb{Z}/p\mathbb{Z}$”, E. Balandraud, Israel Journal of Mathematics, 188 (2012), 405-429.