The Annals of Statistics

On the existence of saturated and nearly saturated asymmetrical orthogonal arrays

Rahul Mukerjee and C. F. Jeff Wu

Full-text: Open access

Abstract

We develop a combinatorial condition necessary for the existence of a saturated asymmetrical orthogonal array of strength 2. This condition limits the choice of integral solutions to the system of equations in the Bose-Bush approach and can thus strengthen considerably the Bose-Bush approach as applied to a symmetrical part of such an array. As a consequence, several nonexistence results follow for saturated and nearly saturated orthogonal arrays of strength 2. One of these leads to a partial settlement of an issue left open in a paper by Wu, Zhang and Wang. Nonexistence of a class of saturated asymmetrical orthogonal arrays of strength 4 is briefly discussed.

Article information

Source
Ann. Statist., Volume 23, Number 6 (1995), 2102-2115.

Dates
First available in Project Euclid: 15 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034713649

Digital Object Identifier
doi:10.1214/aos/1034713649

Mathematical Reviews number (MathSciNet)
MR1389867

Zentralblatt MATH identifier
0897.62083

Subjects
Primary: 62K15: Factorial designs 05B15: Orthogonal arrays, Latin squares, Room squares

Keywords
Bose-Bush bound Delsarte theory

Citation

Mukerjee, Rahul; Wu, C. F. Jeff. On the existence of saturated and nearly saturated asymmetrical orthogonal arrays. Ann. Statist. 23 (1995), no. 6, 2102--2115. doi:10.1214/aos/1034713649. https://projecteuclid.org/euclid.aos/1034713649


Export citation

References

  • BOSE, R. C. and BUSH, K. A. 1952. Orthogonal array s of strength two and three. Ann. Math. Statist. 23 508 524. Z.
  • CHENG, C. S. 1980. Orthogonal array s with variable numbers of sy mbols. Ann. Statist. 8 447 453. Z. Z Z..
  • DAWSON, J. E. 1985. A construction for generalized Hadamard matrices GH 4q, EA q. J. Statist. Plann. Inference 11 103 110. Z.
  • DE LAUNEY, W. 1986. A survey of generalized Hadamard matrices and difference matrices Z. D k, ; G with large k. Utilitas Math. 30 5 29.
  • DELSARTE, P. 1973. An algebraic approach to the association schemes of coding theory. Phillips Research Reports Supplement 10. Z.
  • HONG, Y. 1986. On the nonexistence of nontrivial perfect e-codes and tight 2e-designs in Z. Hamming schemes H n, q with e 3 and q 3. Graphs Combin. 2 145 164. Z.
  • MACWILLIAMS, F. J. and SLOANE, N. J. A. 1977. The Theory of Error-Correcting Codes. NorthHolland, Amsterdam. Z.
  • MUKERJEE, R. and KAGEy AMA, S. 1994. On existence of two sy mbol complete orthogonal array s. J. Combin. Theory Ser. A 66 176 181. Z.
  • MUKERJEE, R. and WU, C. F. J. 1993. On the existence of saturated asy mmetrical orthogonal array s. Statistics Technical Report 93-08, Univ. Waterloo. Z.
  • NODA, R. 1979. On orthogonal array s of strength 4 achieving Rao's bound. J. London Math. Soc. 19 385 390. Z.
  • RAO, C. R. 1947. Factorial experiments derivable from combinatorial arrangements of array s. J. Roy. Statist. Soc. Ser. B 9 128 139. Z.
  • RAO, C. R. 1973. Some combinatorial problems of array s and applications to design of experiZ. ments. In A Survey of Combinatorial Theory J. N. Srivastava, ed. 349 359. NorthHolland, Amsterdam. Z.
  • SEIDEN, E. 1954. On the problem of construction of orthogonal array s. Ann. Math. Statist. 25 151 156. Z.
  • TAGUCHI, G. 1987. Sy stem of Experimental Design. UNIPUB, White Plains, NY. Z.
  • WANG, J. C. 1989. Orthogonal array s and nearly orthogonal array s with mixed levels: construction and applications. Ph.D. Thesis, Univ. Wisconsin, Madison. Z.
  • WANG, J. C. and WU, C. F. J. 1991. An approach to the construction of asy mmetrical orthogonal array s. J. Amer. Statist. Assoc. 86 450 456. Z.
  • WANG, J. C. and WU, C. F. J. 1992. Nearly orthogonal array s with mixed levels and small runs. Technometrics 34 409 422.Z.
  • WU, C. F. J., ZHANG, R. and WANG, R. 1992. Construction of asy mmetrical orthogonal array s of Z k mZ r1.n1 Z rt.nt. the ty pe OA s, s s s. Statist. Sinica 2 203 219.