New Constructions of Asymptotically Optimal Periodic and Aperiodic Quasi-Complementary Sequence Sets

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code division multiple access (MC-CDMA) systems as they can support more users than perfect complementary sequence sets (PCSSs). The objective of this paper is to present new constructions of asymptotically optimal periodic and aperiodic QCSSs with large set sizes. Firstly, we construct a family of asymptotically optimal periodic (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ⁿ</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pⁿ</i> − 1, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pⁿ</i> − 1, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pⁿ</i> + 1) QCSSs with small alphabet size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>, which has larger set size than the known family of periodic (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup>−1), <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup>−1, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ⁿ</i>−1, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ⁿ</i>+1) QCSSs. Secondly, we construct five new families of asymptotically optimal aperiodic QCSSs with large set sizes and low aperiodic tolerances. Each family of these aperiodic QCSSs has set size Θ(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>) for some flock size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i>. Compared with known asymptotically optimal aperiodic QCSSs in the literature, our proposed aperiodic QCSSs have better or new parameters. Particularly, for three families of the costructed aperiodic QCSSs, the column sequence peak-to-average power ratio (PAPR) is upper bounded by p if we select suitable column orthogonal complex matrices.