  A derivation of the basis of states for the $SM(2,4k)$ superconformal minimal
models is presented. It relies on a general hypothesis concerning the role of
the null field of dimension $2k-1/2$. The basis is expressed solely in terms of
$G_r$ modes and it takes the form of simple exclusion conditions (being thus a
quasi-particle-type basis). Its elements are in correspondence with
$(2k-1)$-restricted jagged partitions. The generating functions of the latter
provide novel fermionic forms for the characters of the irreducible
representations in both Ramond and Neveu-Schwarz sectors.
