A new Transformation for Costas Arrays

Abstract

A Costas array of size n is an n × n binary matrix such that no two of the $\binom{n}{2}$ line segments connecting 1s have the same length and slope. Costas arrays are found by finite-field-based construction methods and their manipulations (systematically constructed) and exhaustive search methods. The arrays found exhaustively, which are of completely unknown origin, are called sporadic. Most studies in Costas arrays have tended to focus on systematically constructed Costas arrays rather than sporadic ones, which reveals the hardness of examining a link between systematically constructed Costas arrays and sporadic ones. This paper introduces a new transformation that preserves the Costas property for some Costas arrays, but not all. We observed that this transformation could transform some systematically constructed Costas arrays to sporadic ones and vice versa. Moreover, we introduce a family of arrays with the property that the autocorrelation of each array and the cross-correlation between any two arrays in this family is bounded above by two.