Highly configurable software systems are crucial in practice to satisfy the rising demand for software customization, and combinatorial interaction testing (CIT) is an important methodology for testing such systems. Constrained covering array generation (CCAG), as the core problem in CIT, is to construct a t-wise covering array (CA) of minimum size, where t represents the testing strength. Extensive studies have demonstrated that high-strength CIT (e.g., 4-wise and 5-wise CIT) has stronger fault detection capability than low-strength CIT (i.e., 2-wise and 3-wise CIT), and there exist certain critical faults that can be disclosed through high-strength CIT. Although existing CCAG algorithm has exhibited effectiveness in solving the low-strength CCAG problem, they suffer the severe high-strength challenge when solving 4-wise and 5-wise CCAG, which urgently calls for effective solutions to solving 4-wise and 5-wise CCAG problems. To alleviate the high-strength challenge, we propose a novel and effective local search algorithm dubbed HSCA. Particularly, HSCA incorporates three new and powerful techniques, i.e., multi-round CA generation mechanism, dynamic priority assigning technique, and variable grouping strategy, to improve its performance. Extensive experiments on 35 real-world and synthetic instances demonstrate that HSCA can generate significantly smaller 4-wise and 5-wise CAs than existing state-of-the-art CCAG algorithms. More encouragingly, among all 35 instances, HSCA successfully builds 4-wise and 5-wise CAs for 35 and 29 instances, respectively, including 11 and 15 instances where existing CCAG algorithms fail. Our results indicate that HSCA can effectively mitigate the high-strength challenge.