Line-parallelisms of PG$(n,2)$ from Preparata-like codes

Partitions of the binary linear Hamming code into Preparata-like codes are known to induce line-parallelisms of PG$(n,2)$. In this paper, we show that if $P$ is any Preparata-like code contained in the binary linear Hamming code $H$ of the same length, then $H$ can be partitioned into additive translates of $P$. This generalizes a result of Baker, van Lint, and Wilson who prove this fact for the class of generalized Preparata codes. We give an explicit description for line-parallelisms obtained from such a partition via crooked Preparata-like codes and establish an equivalence criterion for such line-parallelisms.