An Algorithm for constructing Hjelmslev planes

TL;DR

This paper introduces an algorithm for constructing projective and affine Hjelmslev planes using projective planes, affine planes, and orthogonal arrays, revealing structural relationships among these geometries.

Contribution

It provides a novel construction method for 2-uniform Hjelmslev planes and demonstrates their sub-geometry relationships.

Findings

All 2-uniform projective Hjelmslev planes can be constructed with the algorithm.

All 2-uniform affine Hjelmslev planes are sub-geometries of 2-uniform projective Hjelmslev planes.

The construction links Hjelmslev planes to classical geometries and orthogonal arrays.

Abstract

Projective Hjelmslev planes and Affine Hjelmselv planes are generalisations of projective planes and affine planes. We present an algorithm for constructing a projective Hjelmslev planes and affine Hjelsmelv planes using projective planes, affine planes and orthogonal arrays. We show that all 2-uniform projective Hjelmslev planes, and all 2-uniform affine Hjelsmelv planes can be constructed in this way. As a corollary it is shown that all 2-uniform Affine Hjelmselv planes are sub-geometries of 2-uniform projective Hjelmselv planes.