Modified Jacobi sequences

Abstract

Quadratic residue and twin prime sequences are well known types of binary sequence with ideal periodic autocorrelation functions. These are just special cases of much larger families of sequences referred to here as Legendre sequences and modified Jacobi sequences. Although these more general forms are suboptimal and do not have ideal autocorrelation functions, nevertheless they do exhibit out-of-phase autocorrelation values which are independent of their lengths, and so the longer sequences may be useful. The structure and properties of these sequences are investigated, and a two-dimensional array representation that enables the sequences and their autocorrelation functions to be derived in a simple and compact manner are employed .