A Family of Polyphase Sequences With Asymptotically Optimal Correlation

Abstract

Sequences with low correlation have important applications in communications, radar, and cryptography. In this paper, a simple construction of polyphase sequences using additive and multiplicative characters over the finite field ${\mathrm {F}}_{q}$ is proposed. The construction works for any finite field ${\mathrm {F}}_{q}$ with $q>2$ and generates a family of $q-1$ sequences with period $q-1$ and maximum correlation $\sqrt {q}$ . This family is asymptotically optimal with respect to the well-known Welch bound. Most notably, the maximum autocorrelation magnitude of each sequence in this family is equal to 1, and every two distinct sequences are orthogonal to each other. The distribution of the correlation magnitudes of this family is also established.