  We give some examples of Calabi-Yau 3-folds with $\rho=1$, defined over
$\mathbb{Q}$ and constructed as 4-codimensional subvarieties of $\mathbb{P}^7$
via commutative algebra methods. We explain how to deduce their Hodge diamond
and top Chern classes from computer based computations over some finite field
$\mathbb{F}_{p}$. Three of our examples are new. These examples are built out
of Gulliksen-Neg\r{a}rd and Kustin-Miller complexes of locally free sheaves.
Finally, we give two new examples of Calabi-Yau 3-folds of $\mathbb{P}^6$ of
degree 14 and 15 that are not deformation-equivalent to the previously known
examples of these degrees in $\mathbb{P}^6$ (even though they share the same
invariants $(H^3, c_2\cdot H, c_3)$ and $\rho=1$).
