Study of construction of Golomb Costas arrays with ideal autocorrelation properties based on extension field

This paper proposes a specific algebraic structure and demonstrates its nature as an extension field, enabling the construction of Golomb Costas (GC) arrays. It provides detailed instructions and examples for constructing GC arrays using this extension field, along with a corresponding flowchart. Additionally, the paper conducts a thorough analysis, incorporating calculations and comparisons, to evaluate the autocorrelation of a GC array derived from the extension field compared to that of a diagonal frequency hopping array. The analysis reveals the superior autocorrelation properties of GC arrays based on the extension field. Furthermore, the paper establishes a mathematical model for the signal coded by the frequency hopping array and subsequently simulates and compares the ambiguity function of the signal coded by a GC array with that of a signal coded by a diagonal frequency hopping array. This comparison underscores the thumbtack ambiguity function of frequency hopping signal coded by a GC array. Moreover, the paper thoroughly investigates the relationship between the correlation function of GC arrays and the roots of an algebraic equation in a finite field, and strictly proves the ideal autocorrelation properties of Golomb Costas arrays.