Signed magic rectangles with two filled cells in each column

TL;DR

This paper characterizes the existence of signed magic rectangles with two filled cells per column, establishing precise conditions based on the parameters m, n, and r.

Contribution

It provides a complete characterization of when signed magic rectangles with two filled cells per column exist, filling a gap in combinatorial design theory.

Findings

Existence when m=2 and n=r≡0,3 mod 4

Existence for m,r≥3 with mr=2n

Non-existence outside these conditions

Abstract

A signed magic rectangle $SMR(m,n;r, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (ms-1)/2\}$ if $mr$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mr/2\}$ if $mr$ is even, such that precisely $r$ cells in every row and $s$ cells in every column are filled, every integer from set $X$ appears exactly once in the array and the sum of each row and of each column is zero. In this paper we prove that a signed magic rectangle $SMR(m,n;r, 2)$ exists if and only if either $m=2$ and $n=r\equiv 0,3 \pmod 4$ or $m,r\geq 3$ and $mr=2n$.