Some Notes on Prime-Square Sequences

The well-known binary Legendre sequences possess good autocorrelation functions and high linear complexity, and are just special cases of much larger families of cyclotomic sequences. Prime-square sequences are the generalization of these Legendre sequences, but the ratio of the linear complexity to the least period of these sequences approximates to zero if the prime is infinite. However, a relatively straightforward modification can radically improve this situation. The structure and properties, including linear complexity, minimal polynomial, and autocorrelation function, of these modified prime-square sequences are investigated. The hardware implementation is also considered.