Algebraic symmetries of generic $(m+1)$ dimensional periodic Costas arrays

TL;DR

This paper identifies two generators for the symmetry group of generic $(m+1)$-dimensional periodic Costas arrays over elementary abelian groups, and conjectures these generate the entire symmetry group.

Contribution

It introduces two specific generators for the symmetry group of these arrays and conjectures they fully characterize the group.

Findings

Two generators are sufficient to describe the symmetry group in computed examples.

Exhaustive search supports the conjecture that these generators characterize the entire symmetry group.

The work extends understanding of symmetries in high-dimensional Costas arrays.

Abstract

In this work we present two generators for the group of symmetries of the generic $(m+1)$ dimensional periodic Costas arrays over elementary abelian $(\mathbb{Z}_p)^m$ groups: one that is defined by multiplication on $m$ dimensions and the other by shear (addition) on $m$ dimensions. Through exhaustive search we observe that these two generators characterize the group of symmetries for the examples we were able to compute. Following the results, we conjecture that these generators characterize the group of symmetries of the generic $(m+1)$ dimensional periodic Costas arrays over elementary abelian $(\mathbb{Z}_p)^m$ groups.