Unlocking the Strength of the Combined Measure via Correlation

This paper studies the pseudorandomness of four celebrated binary sequence constructions by several authors (Goubin, Gyarmati, Mauduit, Pethő, Rivat, and Sárközy) based on computational evidence. Furthermore, a new theorem establishes a strong link between the correlation measure and the combined measure: for families of sequences satisfying a certain closure property: low correlation (𝐶ℓ) implies a low combined measure (𝑄ℓ). This theoretical result confirms that controlling standard correlation is a powerful method for controlling correlation along arithmetic progressions for broad classes of pseudorandom sequence families. By applying this new theorem, we derive new upper bounds for the combined measure 𝑄ℓ for the established four constructions, thereby confirming their strong combined-measure properties.