Existence of line-parallelisms in finite projective spaces

Determine whether line-parallelisms exist in [?] PG(n,q) for any prime-power q and odd positive integer n.

Background

A line-parallelism of PG(n,q) is a partition of all lines into line-spreads, where each line-spread partitions the points into lines. Although line-spreads are known to exist exactly when n is odd, the general existence problem for line-parallelisms remains unresolved across prime-power orders and odd dimensions.

The paper restricts its constructions to the binary case q=2 and develops new line-parallelisms from partitions of binary linear Hamming codes into translates of Preparata-like codes. These results do not settle the broader existence question for arbitrary prime powers q and odd positive integers n.