There is (almost) no information available on the literature about complex algebraic surfaces of general type with geometric genus $p_g=0,$ self-intersection of the canonical divisor $K^2=3$ and with $5$-torsion. If $S$ is a quintic surface in $\mathbb P^3$ having $15$ $3$-divisible ordinary cusps as only singularities, then there is a Galois triple cover $\phi:X\to S$ branched only at the cusps such that $X$ is regular, $p_g(X)=4,$ $K_X^2=15$ and $\phi$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb Z_5$, so that $X/{\mathbb Z_5}$ is a smooth minimal surface of general type with $p_g=0$ and $K^2=3$. We find two different examples, one of them is the Van der Geer-Zagier's quintic, the other is new.