Constructions of Binary Cross Z-Complementary Pairs With Large CZC Ratio

Abstract

Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase position and end-shift position, also having zero cross-correlation sums around the end-shift position. It can be utilized as a key component in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio $({\mathrm{CZC}}_{\mathrm{ratio}})$ by exploring two promising approaches. The first one of CZCPs via properly cascading sequences from a Golay complementary pair (GCP). The proposed construction leads to $(28L,13L)-\mathrm{CZCPs}$, $(28L,13L+\frac{L}{2})-\mathrm{CZCPs}$ and $(30L,13L-1)-\mathrm{CZCPs}$, where $L$ is the length of a binary GCP. Besides, we emphasize that, our proposed CZCPs have the largest ${\mathrm{CZC}}_{\mathrm{ratio}}=\frac{27}{28}$, compared with known CZCPs but no-perfect CZCPs in the literature. Specially, we proposed optimal binary CZCPs with $(28,13)-\mathrm{CZCP}$ and $(56,27)-\mathrm{CZCP}$. The second one of CZCPs based on Boolean functions (BFs), and the construction of CZCPs have the largest ${\mathrm{CZC}}_{\mathrm{ratio}}=\frac{13}{14}$, compared with known CZCPs but no-perfect CZCPs in the literature.