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 $f(q)$?



The maximal subset of a finite field where the sum of any subset is non-zero
I could trace this question down to the paper of Erdős and Heilbronn "[On the addition of residue classes mod $p$](https://doi.org/10.4064/aa-9-2-149-159)" (Acta Arith. 17 (1970), 227–229), where it is shown that for a prime $p$, if $A$ is a subset of the $p$-element group with $\lvert A\rvert>6\sqrt{3p}$, then the subset sum set of $A$ is the whole group.


It seems that the first to prove that $\lvert A\rvert>c\sqrt{\lvert G\rvert}$, with an absolute constant $c$, suffices to represent $0$ in any abelian group $G$, was Szemerédi ("[On a conjecture of Erdős and Heilbronn](https://doi.org/10.4064/aa-17-3-227-229)", Acta Arith. 17 (1970), 227–229).


Some tightly related problems are considered in a paper of Eggleton and Erdős
"[Two combinatorial
problems in group theory](https://old.renyi.hu/~p_erdos/1972-25.pdf)" (Acta Arith. 21 (1972), 111–116).


Three years later, Olson ("[Sums of sets of group elements](https://doi.org/10.4064/aa-28-2-147-156)", Acta Arith. 28
(1975), 147–156) proved that if $A$ is a zero-sum-free set in a finite
abelian group $G$, then $\lvert A\rvert<3\sqrt{\lvert G\rvert}$.


Hamidoune and Zémor ("[On zero-free subset sums](https://doi.org/10.4064/aa-78-2-143-152)", Acta Arith. 78 (1996), 143–152) showed that for the group of prime order $p$ it suffices to have $\lvert A\rvert>(1+o(1))\sqrt{2p}$.


The exact result for the prime-order groups was obtained by Nguyen,
Szemerédi, and Vu ("[Subset sums modulo a prime](https://doi.org/10.4064/aa131-4-1)", Acta Arith. 131
(2008), no. 4, 303–316).


There were several further improvements in the general case, with the current
record $\lvert A\rvert<\sqrt{6\lvert G\rvert}$ due to Gao, Huang, Hui, Li, Liu, and Peng ("[Sums
of sets of abelian group elements](https://doi.org/10.1016/j.jnt.2019.07.026)", J. Number Theory, 208 (2020),
208–229). In fact, they prove a stronger result: if $A$ is a finite
zero-sum-free subset of an abelian group, then $A$ spans at least as many as
$1+\lvert A\rvert^2/6$ pairwise distinct subset sums.


Two other important papers on this subject are mentioned in the [answer](https://mathoverflow.net/a/417687) of
[Le p'tit bonhomme](https://mathoverflow.net/users/171221/le-ptit-bonhomme).


