Shiftable Heffter spaces

TL;DR

This paper introduces the concept of shiftable Heffter spaces, providing recursive and direct constructions to generate infinitely many such spaces with specific parameters, expanding the combinatorial design framework.

Contribution

It generalizes shiftable Heffter arrays to shiftable Heffter spaces and offers new recursive and direct construction methods for these spaces.

Findings

Recursive construction yields infinitely many shiftable Heffter spaces.

Direct construction uses pandiagonal magic squares for specific parameters.

Constructs shiftable Heffter spaces for all positive integer triples with m ≥ n.

Abstract

The shiftable Heffter arrays are naturally generalized to the shiftable Heffter spaces. We present a recursive construction which starting from a single shiftable Heffter space leads to infinitely many other shiftable Heffter spaces of the same degree. We also present a direct construction making use of pandiagonal magic squares leading to a shiftable $(16\ell^2,4l;3)$ Heffter space for any $\ell \geq 1$. Combining these constructions we obtain a shiftable $(16\ell^2mn, 4\ell n; 3)$ Heffter space for every triple of positive integers $(\ell,m,n)$ with $m \geq n$.