Perfect Gaussian Integer Sequences Embedding Pre-Given Gaussian Integers

Abstract

A pre-given Gaussian integer (GI) is a GI that is determined before a sequence is designed, and a sequence embedding a pre-given GI is a sequence that contains the GI as part of its components. In this letter, for an arbitrary pre-given GI, we present two constructions that produce perfect GI sequences (PGISs) embedding the pre-given GI with different embedment frequencies. Typically, for an arbitrary even integer <inline-formula><tex-math notation= LaTeX >$N$</tex-math></inline-formula> (<inline-formula><tex-math notation= LaTeX >$N\\geq 4$</tex-math></inline-formula>) and arbitrary pre-given GI <inline-formula><tex-math notation= LaTeX >$c$</tex-math></inline-formula>, one of our constructions can yield a PGIS of period <inline-formula><tex-math notation= LaTeX >$N$</tex-math></inline-formula> and degree 3 that embeds the pre-given GI <inline-formula><tex-math notation= LaTeX >$c$</tex-math></inline-formula> <inline-formula><tex-math notation= LaTeX >$N-2$</tex-math></inline-formula> times. Our constructions provide a high degree of freedom and flexibility for PGIS designs to satisfy the requirements of sequence designs and applications.