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Heffter arrays over partial loops

Raúl M. Falcón, Lorenzo Mella

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 $A$.

Heffter arrays over partial loops

Abstract

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

Paper Structure

This paper contains 11 sections, 9 theorems, 41 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Affine designs
  4. Partially filled arrays
  5. Partial loops
  6. Sum polynomials
  7. A family of sum polynomials describing zero sums from non-associative loops
  8. The basic construction
  9. The general construction
  10. Heffter arrays over partial loops
  11. Conclusion and further works

Key Result

Theorem 1

ADDYCDDYDW There exists an $H(n; k)$ if and only if $3 \leq k \leq n$.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Example 3
  • Example 4
  • Lemma 5
  • Example 6
  • Lemma 7
  • proof
  • Proposition 8
  • Example 9
  • ...and 15 more