Heffter arrays over partial loops

TL;DR

This paper generalizes Heffter arrays to partial loops using block-sum polynomials over affine 1-designs, expanding the combinatorial structures and potential applications in design theory.

Contribution

It introduces the concept of fter arrays over partial loops with block-sum polynomials, broadening the scope of combinatorial array constructions.

Findings

Defined fter arrays over partial loops.

Established properties of fter arrays with block-sum polynomials.

Explored potential applications in combinatorial design theory.

Abstract

A Heffter array over an additive group $G$ is any partially filled array $A$ satisfying that: (1) each one of its rows and columns sum to zero in $G$, and (2) if $i\in G\setminus\{0\}$, then either $i$ or $-i$ appears exactly once in $A$. In this paper, this notion is naturally generalized to that of $\mathcal{B}$-Heffter array over a partial loop, where $\mathcal{B}$ is a set of block-sum polynomials over an affine $1$-design on the set of entries in $A$.