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

## TL;DR

This paper explores how partitions of binary Hamming codes into Preparata-like codes induce line-parallelisms in projective geometry PG(n, 2), generalizing previous results and providing explicit descriptions and criteria for these structures.

## Contribution

It generalizes the partitioning of Hamming codes into Preparata-like codes and describes the resulting line-parallelisms explicitly, extending prior work on generalized Preparata codes.

## Key findings

- Partition of Hamming code into Preparata-like codes induces line-parallelisms.
- Explicit description of line-parallelisms from crooked Preparata-like codes.
- Established an equivalence criterion for these line-parallelisms.

## Abstract

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.