Space-filling designs with superior low-dimensional properties are highly required in computer experiments. Strong orthogonal arrays (SOAs) represent a class of such designs that outperform ordinary orthogonal arrays in their stratification properties within low dimensions. Nevertheless, current methods for constructing high-strength SOAs are rare, and they typically rely on regular designs, thereby limiting the number of runs in the final arrays to prime powers. This study presents new construction methods for three types of SOAs: SOAs of strength three, column-orthogonal SOAs (OSOAs) of strength three and three minus. The resulting designs have run sizes of twice an odd prime power without replications, filling the gaps in run sizes left by existing constructions. The projection properties of Addelman–Kempthorne orthogonal arrays are instrumental in the development of these construction methods.

References

• Addelman, S., & Kempthorne, O. (1961). Some main-effect plans and orthogonal arrays of strength two. The Annals of Mathematical Statistics, 32(4), 1167–1176.
• Bao, J., Ji, L., & Pan, Y. (2023). Constructions of column-orthogonal strong orthogonal arrays (in Chinese). SCIENTIA SINICA Mathematica, 53(2), 233–248. https://doi.org/10.1360/SSM-2022-0062
• Bingham, D., Sitter, R. R., & Tang, B. (2009). Orthogonal and nearly orthogonal designs for computer experiments. Biometrika, 96(1), 51–65. https://doi.org/10.1093/biomet/asn057
• Chen, G., & Tang, B. (2022). Using nonregular designs to generate space-filling designs. Journal of Statistical Planning and Inference, 221, 201–211. https://doi.org/10.1016/j.jspi.2022.04.007
• Cheng, C. S., He, Y., & Tang, B. (2021). Minimal second order saturated designs and their applications to space-filling designs. Statistica Sinica, 31(2), 867–890.
• Fang, K. T., Li, R., & Sudjianto, A. (2006). Design and Modeling for Computer Experiments. Chapman & Hall/CRC.
• He, Y., Cheng, C. S., & Tang, B. (2018). Strong orthogonal arrays of strength two plus. The Annals of Statistics, 46(2), 457–468. https://doi.org/10.1214/17-AOS1555
• He, Y., Lin, C. D., & Sun, F. (2022). A new and flexible design construction for orthogonal arrays for modern applications. The Annals of Statistics, 50(3), 1473–1489. https://doi.org/10.1214/21-AOS2159
• He, Y., & Tang, B. (2013). Strong orthogonal arrays and associated Latin hypercubes for computer experiments. Biometrika, 100(1), 254–260. https://doi.org/10.1093/biomet/ass065
• He, Y., & Tang, B. (2014). A characterization of strong orthogonal arrays of strength three. The Annals of Statistics, 42(4), 1347–1360. https://doi.org/10.1214/14-AOS1225
• Hedayat, A. S., Seiden, E., & Stufken, J. (1997). On the maximal number of factors and the enumeration of 3-symbol orthogonal arrays of strength 3 and index 2. Journal of Statistical Planning and Inference, 58(1), 43–63.
• Hedayat, A. S., Sloane, N. J. A., & Stufken, J. (1999). Orthogonal Arrays: Theory and Applications. Springer.
• Jiang, B., Wang, Y., & Sun, F. (2025). On construction of strong orthogonal arrays and column-orthogonal strong orthogonal arrays of strength two plus. Science China Mathematics, 68(5), 1219–1242. https://doi.org/10.1007/s11425-023-2300-6
• Jiang, B., Wang, Z., & Wang, Y. (2021). Strong orthogonal arrays of strength two-plus based on the Addelman-Kempthorne method. Statistics & Probability Letters, 175, 109114. https://doi.org/10.1016/j.spl.2021.109114
• Li, W., Liu, M.-Q., & Yang, J.-F. (2022). Construction of column-orthogonal strong orthogonal arrays. Statistical Papers, 63(2), 515–530. https://doi.org/10.1007/s00362-021-01249-w
• Liu, H., & Liu, M. (2015). Column-orthogonal strong orthogonal arrays and sliced strong orthogonal arrays. Statistica Sinica, 25(4), 1713–1734.
• Santner, T. J., Williams, B. J., & Notz, W. I. (2003). The Design and Analysis of Computer Experiments. Springer.
• Shi, C., & Tang, B. (2019). Design selection for strong orthogonal arrays. Canadian Journal of Statistics, 47(2), 302–314. https://doi.org/10.1002/cjs.v47.2
• Wang, C., Yang, J., & Liu, M. (2021). Construction of strong group-orthogonal arrays. Statistica Sinica, 32(3), 1225–1243.
• Zhou, Y., & Tang, B. (2019). Column-orthogonal strong orthogonal arrays of strength two plus and three minus. Biometrika, 106(4), 997–1004. https://doi.org/10.1093/biomet/asz043

To cite this article: Qiang Gao, Bochuan Jiang, Linyue Shang & Yaping Wang (2026) Construction of strong orthogonal arrays of strength three and three minus via Addelman Kempthorne orthogonal arrays, Statistical Theory and Related Fields, 10:1, 135-153, DOI: 10.1080/24754269.2026.2616871 To link to this article: https://doi.org/10.1080/24754269.2026.2616871