  Using the coupled cluster method (CCM) we study the full (zero-temperature)
ground-state (GS) phase diagram of a spin-half ($s=1/2$) $J_{1}$-$J_{2}$
Heisenberg model on a cross-striped square lattice. Each site of the square
lattice has 4 nearest-neighbour exchange bonds of strength $J_{1}$ and 2
next-nearest-neighbour (diagonal) bonds of strength $J_{2}$. The $J_{2}$ bonds
are arranged so that the basic square plaquettes in alternating columns have
either both or no $J_{2}$ bonds included. The classical ($s \rightarrow
\infty$) version of the model has 4 collinear phases when $J_{1}$ and $J_{2}$
can take either sign. Three phases are antiferromagnetic (AFM), showing
so-called N\'{e}el, double N\'{e}el and double columnar striped order
respectively, while the fourth is ferromagnetic. For the quantum $s=1/2$ model
we use the 3 classical AFM phases as CCM reference states, on top of which the
multispin-flip configurations arising from quantum fluctuations are
incorporated in a systematic truncation hierarchy. Calculations of the
corresponding GS energy, magnetic order parameter and the susceptibilities of
the states to various forms of valence-bond crystalline (VBC) order are thus
carried out numerically to high orders of approximation and then extrapolated
to the (exact) physical limit. We find that the $s=1/2$ model has 5 phases,
which correspond to the four classical phases plus a new quantum phase with
plaquette VBC order. The positions of the 5 quantum critical points are
determined with high accuracy. While all 4 phase transitions in the classical
model are first order, we find strong evidence that 3 of the 5 quantum phase
transitions in the $s=1/2$ model are of continuous deconfined type.
