Improved arithmetic for generating Costas arrays using welch construction method

Costas arrays are widely used in radar waveform design, especially for frequency-coding signal design. Among all the construction methods, exhaustion method requires many calculations, making it time-consuming to generate Costas arrays with large orders. Many algebraic construction methods for Costas arrays, which require fewer calculations, are developed. Among these methods, Welch construction method is easy to achieve and thus widely used. However, Welch construction method is ineffective for Costas arrays with certain orders on system based on IEEE standard, such as 40 and 42, which should be effective in theory. Through conducting further researches that indicate Welch construction method is ineffective due to the exponential increase of intermediate variables, this study discusses this problem and develops a new arithmetic to solve it.