Separating symmetric polynomials over finite fields

The set $S(n)$ of all elementary symmetric polynomials in $n$ variables is a minimal generating set for the algebra of symmetric polynomials in $n$ variables, but over a finite field ${\mathbb{F}}_q$ the set $S(n)$ is not a minimal separating set for symmetric polynomials in general. We determined when $S(n)$ is a minimal separating set for the algebra of symmetric polynomials having the least possible number of elements.