Linear Type-p Most-Perfect Squares

TL;DR

This paper introduces type-p most-perfect squares, a generalization of magic squares, and presents a linear construction method for prime-power orders that could impact their enumeration and related square constructions.

Contribution

It provides a novel linear construction for type-p most-perfect squares in prime-power orders, expanding the understanding and generation of these combinatorial objects.

Findings

Linear construction method for type-p most-perfect squares

Application to generalized Franklin squares

Potential implications for counting such squares

Abstract

We describe a generalization of most-perfect magic squares, called type-p most-perfect squares, and in prime-power orders we give a linear construction of these squares reminiscent of de la Loubere's classical magic square construction method. Type-p most-perfect squares can be used to construct other interesting squares (e.g., generalized Franklin squares) and our linear construction may have implications for counting type-p most-perfect squares.