Costas cubes

2017/02/17 by Jonathan Jedwab, Jedwab, Jonathan, Lily Yen +1 · 2 citations
Economics, Econometrics and Finance · #05B20 #Combinatorics (math.CO) #Economic Theory and Policy #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)

paper · pdf · doi:10.48550/arxiv.1702.05473

openalex publication_date 2017/02/17 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

A Costas array is a permutation array for which the vectors joining pairs of 1s are all distinct. We propose a new three-dimensional combinatorial object related to Costas arrays: an order n Costas cube is an array (d_i,j,k) of size n × n × n over ℤ₂ for which each of the three projections of the array onto two dimensions, namely (∑_i d_i,j,k) and (∑_j d_i,j,k) and (∑_k d_i,j,k), is an order n Costas array. We determine all Costas cubes of order at most 29, showing that Costas cubes exist for all these orders except 18 and 19 and that a significant proportion of the Costas arrays of certain orders occur as projections of Costas cubes. We then present constructions for four infinite families of Costas cubes.

Cited by

• Three-dimensional symmetric designs of propriety 3
• On the cross-correlation properties of large-size families of Costas arrays

Related

• Packing Costas Arrays
• Multidimensional Costas Arrays and Their Periodicity
• On the generalization of the Costas property to higher dimensions
• A Recursive Method for Enumeration of Costas Arrays
• Parity properties of Costas arrays defined via finite fields