  This paper is to study vertex operator superalgebras which are strongly
generated by their weight-$2$ and weight-$\frac{3}{2}$ homogeneous subspaces.
Among the main results, it is proved that if such a vertex operator
superalgebra $V$ is simple, then $V_{(2)}$ has a canonical commutative
associative algebra structure equipped with a non-degenerate symmetric
associative bilinear form and $V_{(\frac{3}{2})}$ is naturally a
$V_{(2)}$-module equipped with a $V_{(2)}$-valued symmetric bilinear form and a
non-degenerate ($\mathbb{C}$-valued) symmetric bilinear form, satisfying a set
of conditions. On the other hand, assume that $A$ is any commutative
associative algebra equipped with a non-degenerate symmetric associative
bilinear form and assume that $U$ is an $A$-module equipped with a symmetric
$A$-valued bilinear form and a non-degenerate ($\mathbb{C}$-valued) symmetric
bilinear form, satisfying the corresponding conditions. Then we construct a Lie
superalgebra $\mathcal{L}(A,U)$ and a simple vertex operator superalgebra
$L_{\mathcal{L}(A,U)}(\ell,0)$ for every nonzero number $\ell$ such that
$L_{\mathcal{L}(A,U)}(\ell,0)_{(2)}=A$ and
$L_{\mathcal{L}(A,U)}(\ell,0)_{(\frac{3}{2})}=U$.
