  The Boolean product $R = P \cdot Q$ of two $\{ 0, 1\} \; m \times m \; $
matrices is $$R(j,k) = 1 \; \mathrm{\ IF\ for\ some\ } \; t \; \,P(j, t) = Q(t,
k) = 1\; \; \mathrm{ELSE\ } \, R(j, k) = 0. $$ The near-optimal design reduces
the complexity of computing $R$ from the standard $m^3$ to $O(m^{(2+e)})$, for
arbitrary small $e > 0$, by a practical algorithm. This renders reduced
complexity to several graph-property tests: Finding triangles and higher-size
cliques; finding all-pairs shortest paths, and more. Also, parsing a string $w$
by a context-free grammar is reduced to near quadratic in $w$-size. The design
uses several distinct 2-digit encodings: $j$ by $(j_1, j_2), \; k \, $ by $\,
(k_1, k_2)$. Each one gives rise to bunches of short digraphs from sources
$j$'s to sinks $k$'s via switching nodes, and walks between them. The combined
information, using the Chinese remainder theorem, leads to the correct values
of $R(j, k)$.
