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The Low Earth Orbit (LEO)-based Positioning, Navigation, and Timing (PNT) System has been proposed as an attractive global navigation system because of its strong transmission power and convergence of Precise Point Positioning (PPP). However, constructing a PN family satisfies the constellation size of the LEO-PNT System while maintaining a correlation performance is a major challenge. In this paper, we propose the extended Weil family, constructed by concatenating two Weil sequences based on the Goldbach conjecture. Among 157 prime pairs capable of generating balanced 10,230-chip codes, the prime pair (p=10091,q=139) yields the largest number of candidate codes satisfying the auto-correlation function (ACF) threshold of BeiDou Navigation Satellite System (BDS) B1 Civil (B1C) family. Using this pair, the total of 254 codes satisfy the BDS B1C correlation thresholds, approximately twice the size of the B1C family. We conducted additional experiments with relaxed cross-correlation function (CCF) thresholds to achieve the target family size of 588, providing two codes for each of the 294 satellites. As a result, a total of 608 codes satisfy the ACF threshold of the BDS B1C family and the CCF threshold of the Global Positioning System (GPS) L1 Civil (L1C) Family, which is sufficient to support the LEO-PNT System.