Fractional Factorial Designs and Codes

Abstract

Fractional factorial experiments with minimum aberration are commonly used in practice. In this study the characteristics of two-level fractional factorial experiments, namely word length pattern, resolution, aberration etc. are introduced. By exploring the algebraic structure of two-level fractional factorial designs, the connection between coding theory, especially Hamming codes, and fractional factorial designs is investigated. Two-level fractional factorial designs are constructed from codes (Hamming, binary linear and binary cyclic codes), and are ordered by the minimum aberration criterion. Designs and their corresponding codes are listed in a catalog.

Keywords

• Minimum aberration
• Hamming codes
• Fractional factorial designs
• Coding theory

Kesirli Çok Etkenli Tasarımlar ve Kodlar

Abstract

Kesirli çok etkenli en az sapma tasarımları, uygulamada yaygın olarak kullanılmaktadır. Bu çalışmada, 2-düzeyli kesirli çok etkenli tasarımların; kelime uzunluğu yapısı, çözüm, en az sapma vb. gibi özellikleri tanıtılmıştır. İki-düzeyli kesirli çok etkenli tasarımların cebirsel yapısı araştırılarak; kod teorisi, özellikle Hamming kodları, ile kesirli çok etkenli tasarımlar arasındaki ilişki incelenmiştir. 2-düzeyli kesirli çok etkenli tasarımlar, kodlardan (Hamming kodları, ikili doğrusal ve döngüsel kodlar) yararlanarak oluşturulmuş ve en az sapma ölçütüne göre sıralanmıştır. Tasarımlar ve kod olarak karşılıkları bir katalogda toplanmıştır.

Keywords

• En az sapma
• Hamming kodları
• Kesirli çok etkenli tasarımlar
• Kod teorisi

References

1. Arazi, B., 1988. A Commensense Approach to the Theory of Error Correcting Codes. Computer System Series, MIT Pres.
2. Bilous, R. T., Rees, G. H. J.,2003. An Enumeration of Binary Self-Dual Codes of Length 32. http://www.cs.umanitoba.ca/~vanrees/bil.pdf.
3. Box, G. E. P., Hunter, J. S, 1961. The 2k-p Fractional Factorial Designs Part I. Technometrics, 3(3): 311-351.
4. Box, G. E. P., Hunter, W. G., Hunter, J. J., 1978. Statistics for Experiments. John Wiley & Sons, New York, NY.
5. Brouwer, A. E., Cohen A. M., Neguyen M. V. M, 2003. Fractional Factorial Desingns of Strength 3 and Small Run Size. http://win.tue.nl/amc/pub /cbn.pdf.
6. Chen, H., 1998. Some Projective Properties of Fractional Factorial Designs. Statistics & Probabilitiy Letters, 40,185-188.
7. Clark, J. B., Dean, A. M., 2001. Equivalence of Fractional Factorial Designs. Statistica Sinica, 11, 537-547.
8. Danacıoğlu, N., 2005. Kesirli Çok Etkenli Deneylerde Çözüm ve En Az Sapma Kavramı, HÜ, İstatistik Bölümü, Doktora tezi, Ankara (yayımlanmamış).

9. Dey, A., 1985. Orthogonal Fractional Factorial Designs. New Delhi, Wiley Eastern.
10. Franklin, M. F., 1984. Constructions Tables of Minimum Aberration Designs, Technometrics, 236(3): 225-232.
11. Fries, A., Hunter, W. G., 1980. Minimum Aberration 2k-p designs. Technometrics, 22(4): 601-608.
12. Gulliver, T. A., Bhargava V. K. ,2000. New Linear Codes Over GF(8). Applied Mathematics Letters, 13, 17-19.
13. Halmos, P. R., 1999. Finite-Dimensional Vector Spaces. 7th. Ed., Springer-Verlag.
14. Hamming, R. W., 1950. Error Detecting and Error Correcting Codes. The Bell System Technical Journal, XXVI, 2, 147-160.
15. Hedeyat, A. S., Sloane, N. J. A., Stufken J., 1999. Orthogonal Arrays: Theory and Applications. Springer-Verlag, NY.
16. Huffman, W. C., Pless, V., 2003. Fundemantals of Error Correcting Codes. Cambridge University Pres.
17. Kuş, P., 2002. Hata Düzeltme Kodlaması. Kara Harp Okulu Bilim Dergisi, Kara Harp Okulu Basımevi, 2, 18-34,
18. Lin, C. D., Sitter, R. R., 2008. An Isomorphism Check for Two-Level Fractional Factorail Designs. Journal of Statistical Planning and Inference, 138, 1085-1101.
19. MacWilliams, F. J., Sloane N. J. A., 1977. The Theory of Error-Correcting Codes. North-Holland Mathematical Library, Elsevier Science Publishers.
20. Mathews, K. R., 1991. Linear Algebra. University of Queensland.
21. Nguyen, G. D., 1997. A Polynomial Construction of Perfect Codes. Computers Math. Applic., 33(8): 127-131.
22. Pless, V., 1998. Introduction to the Theory of Error-Correcting Codes. John Wiley & Sons, Inc., Third Edition.
23. Pretzel, O., 1992. Error-Correcting Codes and Finite Fields. Clarendon Pres. Oxford, Student Edition.
24. Purser, M., 1995. Introduction to Error-Correcting Codes. Artech House Inc.
25. Rao, C. R., 1946. On Hypercubes of Strength D and A System of Confounding in Factorial Experiments. Bull. Cal. Math. Soc., 38, 67-78.
26. Sloane, N. J. A., Thompson, J. G., 1983. Cyclic Self-dual Codes. IEEE Trans. Information Theory, 29, 364-366.
27. Sun, D. X., Chen, J., Wu, C. F. J., 1993. A Catalogue of Two-Level and Three-Level Fractional Factorial Designs With Small Runs. Internal. Statist. Rev., 61,131-145.
28. Tang, B., Deng, L. Y., 1999. Minimum G2-Aberration for Nonregular Fractional Factorial Designs. Ann. Statist., 27, 1914-1926.
29. Vanstone, S. A., Oorschot van P. C., 1989. An Introduction to Error-Correcting Codes with Applications. Kluwer Academic Publishers
30. Wiggert, D., 1978. Error-Control Coding and Applications. Artech House.
31. Wu, C. F. J., Chen, Y. Y., 1992. A Graph-Aided Method for Planning Two-Level Experiments when Certain Interactions are Important. Technometrics, 34, 162-175.
32. Xu, H., 2009. Algorithmic Construction of Efficient Fractional Factorial Designs with Large Run Sizes. Technometrics, 52(3): 262-277.