Perfect Gaussian Integer Sequences With Two Cycles

The complex sequences including Gaussian integers have received considerable attention in the past due to their wide applications in communications and cryptosystems. This paper proposes three new base sequences along with six known ones to construct two novel classes of perfect Gaussian integer sequences (PGISs). The first is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">two-cycle PGIS</i>, where each Gaussian integer has an absolute value such that these values in a 2<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>-periodic sequence form two cycles of period <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>. Compared to the conventional PGISs, the computational complexity of determining a two-cycle PGIS is reduced by approximately one-half. The second is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">zero-deletion PGIS</i> (ZDPGIS) when the resulting shorter PGIS is obtained from a long even-length PGIS by deleting zero elements. Experimental results show that the shorter PGISs have flexible lengths including odd and even, most of which are either optimal or almost optimal two-cycle. Such ZDPGISs outperform conventional ones that reduce the computational complexity up to 62.5%.