  Let $\mathcal{F}$ and $\mathcal{G}$ be two $t$-uniform families of subsets
over $[k] = \{1,2,...,k\}$, where $|\mathcal{F}| = |\mathcal{G}|$, and let $C$
be the adjacency matrix of the bipartite graph whose vertices are the subsets
in $\mathcal{F}$ and $\mathcal{G}$, and there is an edge between $A\in
\mathcal{F}$ and $B \in \mathcal{G}$ if and only if $A \cap B \neq \emptyset$.
The pair $(\mathcal{F},\mathcal{G})$ is $q$-almost cross intersecting if every
row and column of $C$ has exactly $q$ zeros.
  We consider $q$-almost cross intersecting pairs that have a circulant
intersection matrix $C_{p,q}$, determined by a column vector with $p > 0$ ones
followed by $q > 0$ zeros. This family of matrices includes the identity matrix
in one extreme, and the adjacency matrix of the bipartite crown graph in the
other extreme.
  We give constructions of pairs $(\mathcal{F},\mathcal{G})$ whose intersection
matrix is $C_{p,q}$, for a wide range of values of the parameters $p$ and $q$,
and in some cases also prove matching upper bounds. Specifically, we prove
results for the following values of the parameters: (1) $1 \leq p \leq 2t-1$
and $1 \leq q \leq k-2t+1$. (2) $2t \leq p \leq t^2$ and any $q> 0$, where $k
\geq p+q$. (3) $p$ that is exponential in $t$, for large enough $k$.
  Using the first result we show that if $k \geq 4t-3$ then $C_{2t-1,k-2t+1}$
is a maximal isolation submatrix of size $k\times k$ in the $0,1$-matrix
$A_{k,t}$, whose rows and columns are labeled by all subsets of size $t$ of
$[k]$, and there is a one in the entry on row $x$ and column $y$ if and only if
subsets $x,y$ intersect.
