Design of Extended Weil Code Families via Goldbach-Based Prime Concatenation for LEO-PNT Systems

Low Earth Orbit (LEO)-based positioning, navigation, and timing (PNT) systems have attracted growing interest owing to their strong transmission power and rapid Precise Point Positioning (PPP) convergence. A key challenge in realizing such systems is constructing a pseudo noise (PN) code family large enough to accommodate hundreds of satellites while maintaining competitive correlation performance. In this study, we propose the Concatenated Weil (C.W.) family, an extended Weil construction that concatenates two Weil sequences whose prime periods sum to the target code length, motivated by the Goldbach conjecture. A two-stage search---auto-correlation screening followed by cross-correlation screening---identifies 608 candidate codes satisfying the correlation thresholds of established modernized Radio Navigation Satellite System (RNSS) families. A subsequent spectral refinement based on the minimum-to-average Power Spectral Density (PSD) ratio removes codes with deep spectral nulls, yielding a final family of 588 balanced codes. Benchmarking against modernized RNSS PN families demonstrates that the proposed family achieves the largest family size while maintaining comparable correlation performance, thereby providing a viable PN solution for large-scale LEO-PNT constellations.