[Paper Review] New complete 2-arcs in the uniform projective Hjelmslev planes over chain rings of order 25

Kiermaier, Michael, Matthias Koch|arXiv (Cornell University)|Mar 10, 2015

Coding theory and cryptography6 references3 citations

TL;DR

This paper presents new complete 2-arcs of size 21 in the uniform projective Hjelmslev plane over the chain ring ℤ₂₅ and size 22 over 𝔽₅[X]/(X²), both exceeding previously known bounds. Using a customized backtracking algorithm with isomorphism filtering and a merging technique to accelerate 2-arc validation, the authors identify these maximal arcs and analyze their geometric structure, challenging prior conjectures about arc size dominance in Hjelmslev geometries.

ABSTRACT

In this paper a 2-arc of size 21 in the projective Hjelmslev plane PHG(2,\Z_25) and a 2-arc of size 22 in PHG(2,\F_5[X]/(X^2)) are given. Both arcs are bigger than the 2-arcs previously known in the respective plane. Furthermore, we will give some information on the geometrical structure of the arcs.

Motivation & Objective

• To determine the maximum possible size of 2-arcs in uniform projective Hjelmslev planes over finite chain rings of order 25.
• To investigate whether non-Galois chain rings can host larger 2-arcs than their Galois ring counterparts, challenging a prior conjecture.
• To develop and apply an efficient computational search method that filters isomorphic configurations early and accelerates 2-arc validation using a merging technique.
• To analyze the geometric structure of the newly discovered 2-arcs and assess their significance in the context of Hjelmslev geometry.

Proposed method

• A depth-first backtracking algorithm is employed to search for 2-arcs in the projective Hjelmslev planes over ℤ₂₅ and 𝔽₅[X]/(X²), with isomorphism reduction applied at the first seven levels of the search tree.
• The algorithm uses a merging technique to avoid redundant 2-arc checks: a set is valid if all its 3-point subsets are valid, enabling efficient forward propagation during search.
• Isomorphism testing is performed only on leaf nodes, with automorphism groups of the planes used to guide the search and reduce redundant computation.
• The search exploits the canonical homomorphism φ: R → 𝔽₅ to project the Hjelmslev geometry onto the classical projective plane PG(2,𝔽₅), using the image of the arc as a filter to restrict the search space.
• The method leverages the homomorphism principle to restrict the operating group to the stabilizer of the projected arc in PGL(3,𝔽₅), significantly reducing computational cost.
• The algorithm is implemented using the Leiterspiel framework for canonical labeling and isomorphism testing, enabling efficient handling of the large automorphism groups of the planes.

Experimental results

Research questions

• RQ1 What is the maximum size of a 2-arc in the uniform projective Hjelmslev plane over the chain ring ℤ₂₅?
• RQ2 Can a 2-arc in a non-Galois Hjelmslev plane exceed the size of the largest known 2-arc in its Galois ring counterpart?
• RQ3 How can computational search techniques be optimized to handle the large automorphism groups and isomorphic copies in Hjelmslev geometries?
• RQ4 What geometric structure characterizes the newly discovered 2-arcs in these Hjelmslev planes?

Key findings

• A complete 2-arc of size 21 was discovered in the projective Hjelmslev plane PHG(2, ℤ₂₅), surpassing the previously known maximum of 20.
• A complete 2-arc of size 22 was discovered in PHG(2, 𝔽₅[X]/(X²)), exceeding the previously known maximum of 18 in this plane.
• The arc in PHG(2, 𝔽₅[X]/(X²)) is the largest known 2-arc in any uniform projective Hjelmslev plane over a non-Galois chain ring of order 25.
• The arc in PHG(2, ℤ₂₅) is the first known 2-arc of size 21 in this plane, indicating that the previous bound was not tight.
• The search results suggest a potential counterexample to the conjecture that Galois ring-based Hjelmslev planes always host larger 2-arcs than non-Galois ones with the same q.
• The geometric structure of the arcs reveals that their point sets are not contained in any single fiber of the projection φ, indicating non-trivial distribution across the residue field geometry.