Signed Magic arrays with certain property

TL;DR

This paper proves the existence of certain signed magic arrays with symmetric properties and specific conditions on their dimensions, expanding the understanding of combinatorial array constructions.

Contribution

It establishes the existence criteria for signed magic arrays with symmetric properties for various even dimensions, filling a gap in combinatorial design theory.

Findings

Existence of SMA(m,n) for all even n ≥ 2 under specified conditions.

Construction methods for SMA arrays with symmetric properties.

Characterization of dimension conditions for array existence.

Abstract

A signed magic array, $SMA(m, n;s,t)$, is an $m \times n$ array with the same number of filled cells $s$ in each row and the same number of filled cells $t$ in each column, filled with a certain set of numbers that is symmetric about the number zero, such that every row and column has a zero sum. We use the notation $SMA(m, n)$ if $m=t$ and $n=s$. In this paper, we prove that for every even number $n\geq 2$ there exists an $SMA(m,n)$ such that the entries $\pm x$ appear in the same row for every $x\in\{1, 2, 3,\ldots, mn/2\}$ if and only if $m\equiv 0, 3(\mod4)$ and $n=2$ or $m\geq 3$ and $n\geq 4$.