Perfect Sequences of Odd Prime Length

Abstract

A sequence is said to be perfect if it has an ideal periodic autocorrelation function. In addition, the degree of a sequence is defined as the number of distinct nonzero elements within each period of the sequence. This study presents a systematic method for constructing perfect sequences (PSs) of odd prime periods, where the general constraint equations for the sequence coefficients are derived, providing a solid theoretical foundation to the construction of PSs. The proposed scheme commences by partitioning a cyclic group $\mathbf {Z}_P=\lbrace 1,2, \ldots,P-1\rbrace$ into $K$ cosets of cardinality $M$ , where $P=K\cdot M+1$ is an odd prime. Based on this partition, the degree-( $K$ + 1) PSs are then constructed. Finally, case studies are presented to illustrate the proposed construction.