  Subspace codes are collections of subspaces of a projective space such that
any two subspaces satisfy a pairwise minimum distance criterion. Recent results
have shown that it is possible to construct optimal $(5,3)$ subspace codes from
pairs of partial spreads in the projective space $\mathrm{PG}(4,q)$ over the
finite field $ \mathbb{F}_q $, termed doubling codes. We have utilized a
complete classification of maximal partial line spreads in $\mathrm{PG}(4,2)$
in literature to establish the types of the spreads in the doubling code
instances obtained from two recent constructions of optimum $(5,3)_q$ codes,
restricted to $ \mathbb{F}_2 $. Further we present a new characterization of a
subclass of binary doubling codes based on the intersection patterns of key
subspaces in the pair of constituent spreads.
