On a Grouping Method for Constructing Mixed Orthogonal Arrays
Author(s)
Chung-Yi Suen
ABSTRACT
Mixed orthogonal arrays of strength two and size smn are constructed by grouping points in the finite projective geometry PG(mn-1, s). PG(mn-1, s) can be partitioned into [(smn-1)/(sn-1)](n-1)-flats such that each (n-1)-flat is associated with a point in PG(m-1, sn). An orthogonal array Lsmn((sn)(smn-)(sn-1) can be constructed by using (smn-1)/( sn-1) points in PG(m-1, sn). A set of (st-1)/(s-1) points in PG(m-1, sn) is called a (t-1)-flat over GF(s) if it is isomorphic to PG(t-1, s). If there exists a (t-1)-flat over GF(s) in PG(m-1, sn), then we can replace the corresponding [(st-1)/(s-1)] sn-level columns in Lsmn((sn)(smn-)(sn-1) by (smn-1)/( sn-1) st -level columns and obtain a mixed orthogonal array. Many new mixed orthogonal arrays can be obtained by this procedure. In this paper, we study methods for finding disjoint (t-1)-flats over GF(s) in PG(m-1, sn) in order to construct more mixed orthogonal arrays of strength two. In particular, if m and n are relatively prime then we can construct an Lsmn((sm)smn-1/sm-1-i(sn-1)/ (s-1)( sn) i(sm-1)/ s-1) for any 0<i<(smn-1)(s-1)/( sm-1)( sn-1) New orthogonal arrays of sizes 256, 512, and 1024 are obtained by using PG(7,2), PG(8,2), and PG(9,2) respectively.
Mixed orthogonal arrays of strength two and size smn are constructed by grouping points in the finite projective geometry PG(mn-1, s). PG(mn-1, s) can be partitioned into [(smn-1)/(sn-1)](n-1)-flats such that each (n-1)-flat is associated with a point in PG(m-1, sn). An orthogonal array Lsmn((sn)(smn-)(sn-1) can be constructed by using (smn-1)/( sn-1) points in PG(m-1, sn). A set of (st-1)/(s-1) points in PG(m-1, sn) is called a (t-1)-flat over GF(s) if it is isomorphic to PG(t-1, s). If there exists a (t-1)-flat over GF(s) in PG(m-1, sn), then we can replace the corresponding [(st-1)/(s-1)] sn-level columns in Lsmn((sn)(smn-)(sn-1) by (smn-1)/( sn-1) st -level columns and obtain a mixed orthogonal array. Many new mixed orthogonal arrays can be obtained by this procedure. In this paper, we study methods for finding disjoint (t-1)-flats over GF(s) in PG(m-1, sn) in order to construct more mixed orthogonal arrays of strength two. In particular, if m and n are relatively prime then we can construct an Lsmn((sm)smn-1/sm-1-i(sn-1)/ (s-1)( sn) i(sm-1)/ s-1) for any 0<i<(smn-1)(s-1)/( sm-1)( sn-1) New orthogonal arrays of sizes 256, 512, and 1024 are obtained by using PG(7,2), PG(8,2), and PG(9,2) respectively.
KEYWORDS
Finite field; Finite projective geometry; (t-1)-flat over GF(s) in PG(m-1, sn ); Geometric orthogonal array; Matrix representation; Minimal polynomial; Orthogonal main-effect plan; Primitive element; Tight.
Finite field; Finite projective geometry; (t-1)-flat over GF(s) in PG(m-1, sn ); Geometric orthogonal array; Matrix representation; Minimal polynomial; Orthogonal main-effect plan; Primitive element; Tight.
Cite this paper
C. Suen, "On a Grouping Method for Constructing Mixed Orthogonal Arrays," Open Journal of Statistics, Vol. 2 No. 2, 2012, pp. 188-197. doi: 10.4236/ojs.2012.22022.
C. Suen, "On a Grouping Method for Constructing Mixed Orthogonal Arrays," Open Journal of Statistics, Vol. 2 No. 2, 2012, pp. 188-197. doi: 10.4236/ojs.2012.22022.
References
[1] R. C. Bose and K. A. Bush, “Orthogonal Arrays of Strength Two and Three,” The Annals of Mathematical Statistics, Vol. 23, No. 4, 1952, pp. 508-524. doi:10.1214/aoms/1177729331 [2] R. L. Plackett and J. P. Burma, “The Design of Optimum Multifactorial Experiments,” Biometrika, Vol. 33, No. 4, 1946, No. 4, pp. 305-325. [3] C. R. Rao, “Factorial Experiments Derivable from Combinatorial Arrangements of Arrays,” Journal of Royal Statistical Society (Supplement), Vol. 9, No. 1, 1947, pp. 128-139. doi:10.2307/2983576 [4] C. R. Rao, “Some Combinatorial Problems of Arrays and Applications to Design of Experiments,” In: J. N. Srivastava, Ed., A Survey of Combinatorial Theory, North- Holland, Amsterdam, 1973, pp. 349-359. [5] J. C. Wang and C. F. J. Wu, “An Approach to the Construction of Asymmetrical Orthogonal Arrays,” Journal of American Statistical Association, Vol. 86, No. 414, 1991, pp. 450-456. doi:10.2307/2290593 [6] C. F. J. Wu, R. C. Zhang and R. Wang, “Construction of Asymmetrical Orthogonal Array of Type OA(sk, sm(sr1)n1 ??? (srt)nt),” Statistica Sinica, Vol. 2, No. 1, 1992, pp. 203- 219. [7] A. Dey and C. K. Midha, “Construction of Some Asymmetrical Orthogonal Arrays,” Statistics & Probability Letters, Vol. 28, No. 3, 1996, pp. 211-217. doi:10.1016/0167-7152(95)00126-3 [8] Y. S. Zhang, Y. Q. Lu and S. Q. Pang, “Orthogonal Arrays Obtained by Orthogonal Decomposition of Projection Matrices,” Statistica Sinica, Vol. 9, No. 2, 1999, pp. 595-604. [9] C. Suen and W. F. Kuhfeld, “On the Construction of Mixed Orthogonal Arrays of Strength Two,” Journal of Statistical Planning and Inference, Vol. 133, No. 2, 2005, pp. 555-560. doi:10.1016/j.jspi.2004.03.018 [10] A. S. Hedayat, N. J. A. Sloane and J. Stufken, “Orthogonal Arrays,” Springer, New York, 1999. doi:10.1007/978-1-4612-1478-6 [11] S. Addelman, “Orthogonal Main Effect Plans for Asymmetrical Factorial Experiments,” Technometrics, Vol. 4, No. 1, 1962, pp. 21-46. doi:10.2307/1266170 [12] C. F. J. Wu, “Construction of 2m4n Deigns via a Grouping Scheme,” Annals of Statistics, Vol. 17, No. 4, 1989, pp. 1880-1885. doi:10.1214/aos/1176347399 [13] E. M. Rains, N. J. A. Sloane and J. Stufken, “The Lattice of N-Run Orthogonal Arrays,” Journal of Statistical Planning and Inference, Vol. 102, No. 2, 2002, pp. 477- 500. doi:10.1016/S0378-3758(01)00119-7 [14] C. Suen, A. Das and A. Dey, “On the Construction of Asymmetric Orthogonal Arrays,” Statistica Sinica, Vol. 11, No. 1, 2001, pp. 241-260. [15] J. W. P. Hirschefld, “Projective Geometries over Finite Fields,” Oxford University Press, Oxford, 1979. [16] C. Suen and A. Dey, “Construction of Asymmetric Orthogonal Arrays through Finite Geometries,” Journal of Statistical Planning and Inference, Vol. 115, No. 2, 2003, pp. 623-635. doi:10.1016/S0378-3758(02)00165-9
[1] R. C. Bose and K. A. Bush, “Orthogonal Arrays of Strength Two and Three,” The Annals of Mathematical Statistics, Vol. 23, No. 4, 1952, pp. 508-524. doi:10.1214/aoms/1177729331 [2] R. L. Plackett and J. P. Burma, “The Design of Optimum Multifactorial Experiments,” Biometrika, Vol. 33, No. 4, 1946, No. 4, pp. 305-325. [3] C. R. Rao, “Factorial Experiments Derivable from Combinatorial Arrangements of Arrays,” Journal of Royal Statistical Society (Supplement), Vol. 9, No. 1, 1947, pp. 128-139. doi:10.2307/2983576 [4] C. R. Rao, “Some Combinatorial Problems of Arrays and Applications to Design of Experiments,” In: J. N. Srivastava, Ed., A Survey of Combinatorial Theory, North- Holland, Amsterdam, 1973, pp. 349-359. [5] J. C. Wang and C. F. J. Wu, “An Approach to the Construction of Asymmetrical Orthogonal Arrays,” Journal of American Statistical Association, Vol. 86, No. 414, 1991, pp. 450-456. doi:10.2307/2290593 [6] C. F. J. Wu, R. C. Zhang and R. Wang, “Construction of Asymmetrical Orthogonal Array of Type OA(sk, sm(sr1)n1 ??? (srt)nt),” Statistica Sinica, Vol. 2, No. 1, 1992, pp. 203- 219. [7] A. Dey and C. K. Midha, “Construction of Some Asymmetrical Orthogonal Arrays,” Statistics & Probability Letters, Vol. 28, No. 3, 1996, pp. 211-217. doi:10.1016/0167-7152(95)00126-3 [8] Y. S. Zhang, Y. Q. Lu and S. Q. Pang, “Orthogonal Arrays Obtained by Orthogonal Decomposition of Projection Matrices,” Statistica Sinica, Vol. 9, No. 2, 1999, pp. 595-604. [9] C. Suen and W. F. Kuhfeld, “On the Construction of Mixed Orthogonal Arrays of Strength Two,” Journal of Statistical Planning and Inference, Vol. 133, No. 2, 2005, pp. 555-560. doi:10.1016/j.jspi.2004.03.018 [10] A. S. Hedayat, N. J. A. Sloane and J. Stufken, “Orthogonal Arrays,” Springer, New York, 1999. doi:10.1007/978-1-4612-1478-6 [11] S. Addelman, “Orthogonal Main Effect Plans for Asymmetrical Factorial Experiments,” Technometrics, Vol. 4, No. 1, 1962, pp. 21-46. doi:10.2307/1266170 [12] C. F. J. Wu, “Construction of 2m4n Deigns via a Grouping Scheme,” Annals of Statistics, Vol. 17, No. 4, 1989, pp. 1880-1885. doi:10.1214/aos/1176347399 [13] E. M. Rains, N. J. A. Sloane and J. Stufken, “The Lattice of N-Run Orthogonal Arrays,” Journal of Statistical Planning and Inference, Vol. 102, No. 2, 2002, pp. 477- 500. doi:10.1016/S0378-3758(01)00119-7 [14] C. Suen, A. Das and A. Dey, “On the Construction of Asymmetric Orthogonal Arrays,” Statistica Sinica, Vol. 11, No. 1, 2001, pp. 241-260. [15] J. W. P. Hirschefld, “Projective Geometries over Finite Fields,” Oxford University Press, Oxford, 1979. [16] C. Suen and A. Dey, “Construction of Asymmetric Orthogonal Arrays through Finite Geometries,” Journal of Statistical Planning and Inference, Vol. 115, No. 2, 2003, pp. 623-635. doi:10.1016/S0378-3758(02)00165-9