The Annals of Statistics

A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions

Runchu Zhang, Frederick K. H. Phoa, Rahul Mukerjee, and Hongquan Xu

Full-text: Open access


The study of good nonregular fractional factorial designs has received significant attention over the last two decades. Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard. The present paper shows how a trigonometric approach can facilitate a systematic understanding of such QC designs and lead to new theoretical results covering hitherto unexplored situations. We focus attention on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal QC designs often have larger generalized resolution and projectivity than comparable regular designs. Moreover, some of these designs are found to have maximum projectivity among all designs.

Article information

Ann. Statist. Volume 39, Number 2 (2011), 931-955.

First available in Project Euclid: 8 April 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K15: Factorial designs

Aliasing index branching technique generalized minimum aberration generalized resolution Gray map nonregular design projectivity


Zhang, Runchu; Phoa, Frederick K. H.; Mukerjee, Rahul; Xu, Hongquan. A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions. Ann. Statist. 39 (2011), no. 2, 931--955. doi:10.1214/10-AOS815.

Export citation


  • [1] Box, G. and Tyssedal, J. (1996). Projective properties of certain orthogonal arrays. Biometrika 83 950–955.
  • [2] Box, G. E. P. and Hunter, J. S. (1961). The 2kp fractional factorial designs. Technometrics 3 311–351. 449-458.
  • [3] Bush, K. A. (1952). Orthogonal arrays of index unity. Ann. Math. Statist. 23 426–434.
  • [4] Butler, N. A. (2003). Minimum aberration construction results for nonregular two-level fractional factorial designs. Biometrika 90 891–898.
  • [5] Chen, H. and Hedayat, A. S. (1996). 2nl designs with weak minimum aberration. Ann. Statist. 24 2536–2548.
  • [6] Chen, J. H. and Wu, C. F. J. (1991). Some results on snk fractional factorial designs with minimum aberration or optimal moments. Ann. Statist. 19 1028–1041.
  • [7] Cheng, C.-S., Deng, L.-Y. and Tang, B. (2002). Generalized minimum aberration and design efficiency for nonregular fractional factorial designs. Statist. Sinica 12 991–1000.
  • [8] Cheng, S.-W., Li, W. and Ye, K. Q. (2004). Blocked nonregular two-level factorial designs. Technometrics 46 269–279.
  • [9] Deng, L.-Y. and Tang, B. (1999). Generalized resolution and minimum aberration criteria for Plackett–Burman and other nonregular factorial designs. Statist. Sinica 9 1071–1082.
  • [10] Fries, A. and Hunter, W. G. (1980). Minimum aberration 2kp designs. Technometrics 22 601–608.
  • [11] Hammons, A. R. Jr., Kumar, P. V., Calderbank, A. R., Sloane, N. J. A. and Solé, P. (1994). The Z4-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40 301–319.
  • [12] Li, Y., Deng, L.-Y. and Tang, B. (2004). Design catalog based on minimum G-aberration. J. Statist. Plann. Inference 124 219–230.
  • [13] Ma, C.-X. and Fang, K.-T. (2001). A note on generalized aberration in factorial designs. Metrika 53 85–93 (electronic).
  • [14] Mukerjee, R. and Wu, C. F. J. (2006). A Modern Theory of Factorial Designs. Springer, New York.
  • [15] Phoa, F. K. H. (2009). Analysis and Construction of Nonregular Fractional Factorial Designs. Ph.D. thesis, Univ. California, Los Angeles.
  • [16] Phoa, F. K. H. and Xu, H. (2009). Quarter-fraction factorial designs constructed via quaternary codes. Ann. Statist. 37 2561–2581.
  • [17] Stufken, J. and Tang, B. (2007). Complete enumeration of two-level orthogonal arrays of strength d with d+2 constraints. Ann. Statist. 35 793–814.
  • [18] Sun, D. X., Li, W. and Ye, K. Q. (2011). An algorithm for sequentially constructing non-isomorphic orthogonal designs and its applications. Unpublished manuscript.
  • [19] Tang, B. (2001). Theory of J-characteristics for fractional factorial designs and projection justification of minimum G2-aberration. Biometrika 88 401–407.
  • [20] Tang, B. and Deng, L.-Y. (1999). Minimum G2-aberration for nonregular fractional factorial designs. Ann. Statist. 27 1914–1926.
  • [21] Tang, B. and Wu, C. F. J. (1996). Characterization of minimum aberration 2nk designs in terms of their complementary designs. Ann. Statist. 24 2549–2559.
  • [22] Wu, C. F. J. and Hamada, M. (2000). Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley, New York.
  • [23] Xu, H. (2005). Some nonregular designs from the Nordstrom–Robinson code and their statistical properties. Biometrika 92 385–397.
  • [24] Xu, H. and Deng, L.-Y. (2005). Moment aberration projection for nonregular fractional factorial designs. Technometrics 47 121–131.
  • [25] Xu, H., Phoa, F. K. H. and Wong, W. K. (2009). Recent developments in nonregular fractional factorial designs. Statist. Surv. 3 18–46.
  • [26] Xu, H. and Wong, A. (2007). Two-level nonregular designs from quaternary linear codes. Statist. Sinica 17 1191–1213.
  • [27] Xu, H. and Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29 1066–1077.
  • [28] Ye, K. Q. (2003). Indicator function and its application in two-level factorial designs. Ann. Statist. 31 984–994.