Fano weighted complete intersections of large codimension

Let $X$ be a smooth Fano variety. The index of $X$ is the largest natural number $i_X$ such that the canonical class $K_X$ is divisible by $i_X$ in the Picard group of $X$. It is well known that $i_X <= n(X) + 1$ for $n(X) = dim(X)$. We are going to consider smooth Fano weighted complete intersections over an algebraically closed field of characteristic zero. It is known that $k(X) <= n(X) + 1 - i_X$ for any such $X$, where $k(X)$ is the codimension of $X$. Let us introduce new invariant $r(X) = n(X) - k(X) - i_X + 1$. In the talk I will outline what is known about smooth Fano weighted complete intersection of given $r(X) = r_0$.