Legendre Sequences and Modified Jacobi Sequences

Abstract

In this paper, a survey of Legendre sequences and modified Jacobi sequences is presented, firstly. We introduce the construction and periodic autocorrelation functions of these two sequences (binary and polyphase). Then we determine the linear complexity of all modified polyphase Jacobi sequences and the corresponding feedback polynomials of the shortest linear feedback shift register that generates such a sequence. Making use of these results, at the same time, we prove the conjectures on the linear complexity and feedback polynomials of modified Jacobi sequences brought forward by D.H. Green and J. Choi.