New Construction of Complementary Sequence (or Array) Sets and Complete Complementary Codes

Abstract

A new method to construct <inline-formula> <tex-math notation= LaTeX >$q$ </tex-math></inline-formula>-ary complementary sequence sets (CSSs) and complete complementary codes (CCCs) of size <inline-formula> <tex-math notation= LaTeX >$N$ </tex-math></inline-formula> is proposed by using desired para-unitary (PU) matrices. The concept of seed PU matrices is introduced and a systematic approach on how to compute the explicit forms of the functions in constructed CSSs and CCCs from the seed PU matrices is given. A general form of these functions only depends on a basis of the functions from <inline-formula> <tex-math notation= LaTeX >${\\mathbb Z}_{N}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation= LaTeX >${\\mathbb Z}_{q}$ </tex-math></inline-formula> and representatives in the equivalent class of Butson-type Hadamard (BH) matrices. Especially, the realization of Golay pairs from our general form exactly coincides with the standard Golay pairs. The realization of ternary complementary sequences of size 3 is first reported here. For the realization of the quaternary complementary sequences of size 4, almost all the sequences derived here are never reported before. Generalized seed PU matrices and the recursive constructions of the desired PU matrices are also studied, and a large number of new constructions of CSSs and CCCs are given accordingly. From the perspective of this paper, all the known results of CSSs and CCCs with explicit GBF form in the literature (except non-standard Golay pairs) are constructed from the WHT matrices. This suggests that the proposed method with other BH matrices will yield a large number of new CSSs and CCCs with the significantly increasing number of the sequences of low peak-to-mean envelope power ratio.