A Systematic Framework for the Construction of Optimal Complete Complementary Codes

Abstract

The complete complementary code (CCC) that was proposed by Suehiro and Hatori is a sequence family, that is a set of sequence sets, with ideal correlation sums. Numerous studies in the literature show its applications to direct-spread code-division multiple access (DS-CDMA) systems for interchannel interference (ICI)-free communication with improved spectral efficiency. In this paper, we propose a systematic framework for the construction of CCCs based on <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -shift cross-orthogonal sequence families ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -CO-SFs) . We show theoretical bounds on the size of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -CO-SFs and CCCs and give a set of four algorithms for their generation and extension. The algorithms are optimal in the sense that the size of the resultant sequence families achieves theoretical bounds and, with the algorithms, we can construct an optimal CCC consisting of sequences whose lengths are not only almost arbitrary but even variable between sequence sets. We also discuss the family size, alphabet size, and length of constructible CCCs based on the proposed algorithms.