A Theorem in Finite Projective Geometry and an Application to Statistics1
نویسندگان
چکیده
R. C. Bose showed that certain statistical problems of design of experiments can be attacked fruitfully by interpreting the statistical terms involved in terms of finite geometries. In particular, the geometrical interpretation proved useful when applied to the problem of determining the maximum number of factors which can be accommodated in symmetrical factorial design without confounding the degrees of freedom belonging to interactions of a given order or lesser. In a paper Mathematical theory of factorial design (Sankhya vol. 8 (1947) pp. 107-166) R. C. Bose proved that the maximum number, mt(r, s), of factors which can be accommodated in symmetrical factorial design in which each factor is at s = p" levels (p being a positive prime and n a positive integer) and each block is of size sr, without confounding any degrees of freedom belonging to any interaction involving t or lesser number of factors, is given by the maximum number of points of a finite projective space PG(r — \, s) (of r homogenous co-ordinates each of which is capable of 5 values) such that no t of the chosen points are conjoint (lie in a space of t — 2 dimensions). Furthermore, R. C. Bose proved in the above mentioned paper that m3(4, s)=j2+l when 5 is a power of an odd prime and s24-l ^m3(4, 5)^52+5+2 when s = 2", n>l. This inequality gives, for 5 = 4, 17gw3(4, 4) ^22. It is the purpose of this paper to prove that m3(4, 4) = 17. In other words: the maximum number of points of PG{3, 4) such that no 3 of them are collinear cannot exceed 17 (Theorem 5). In order to make the reading of this paper independent of the literature on finite geometry we explain shortly the PG(3, 4). The points in PG(3, 4) can be represented by attaching to it a Galois field with 4 elements, GF(4), built with the help of an irreducible polynomial, €24-e4-l =0, where a point is understood to be an ordered quadruple of elements of GF(4), not all zero. Two points (xi, x2, xs, Xa) and (ylt y2, y3, yt) are defined to be identical if there exists a nonzero element of GF(4), sayp, such that x,-=py,-, i— 1, 2,3,4. Thus, the total number of points is seen to be (44 —1)/3 = 85. By a line passing through two different points (xi, x2, x3, x4) and (yi, y2, y%, yi)
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