residual design.

Introduction
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Let ${{\mathbb{F}}}_q$ be the finite field with $q$ elements and let ${{\mathbb{F}}}_q^n$ be the set of all vectors of length $n$ over ${{\mathbb{F}}}_q$. ${{\mathbb{F}}}_q^n$ is a vector space with dimension $n$ over ${{\mathbb{F}}}_q$. For a given integer $k$, $0 \leq k \leq n$, let ${{\cal G}}_q(n,k)$ denote the set of all $k$-dimensional subspaces ($k$-*subspaces* in short) of ${{\mathbb{F}}}_q^n$. ${{\cal G}}_q(n,k)$ is often referred to as a Grassmannian. It is well known that $$\begin{small}
| {{\cal G}}_q (n,k) | = {\nsbinom{{n}}{{k}}_{q}}
{\mbox{$\stackrel{\rm def}{=}$}}\frac{(q^n-1)(q^{n-1}-1) \cdots
(q^{n-k+1}-1)}{(q^k-1)(q^{k-1}-1) \cdots (q-1)}
\end{small}$$ where ${\nsbinom{{n}}{{k}}_{q}}$ is the $q$-*binomial coefficient* (known also as the *Gaussian coefficient* [@vLWi92 pp. 325-332]).

Let $Q$ be a set with $n$ elements. A *$t$-$(n,k,\lambda)$ design*, is a collection of $k$-subsets of $V$, called *blocks*, such that each $t$-subset of $Q$ is contained in exactly $\lambda$ blocks. A $t$-$(n,k,\lambda)$ design with $t = \lambda = 1$ is trivial: it is simply a partition of $Q$ into $k$-subsets, which exists if and only if $k$ divides $n$. A $t$-$(n,k,1)$ design with $t \:{\geq}\, 2$ is known as a *Steiner system*, and usually denoted $S(t,k,n)$. Steiner systems are among the most beautiful and well-studied structures in combinatorics. Their history goes back 