EXISTENCE AND CONSTRUCTION ON ROOM SQUARES BY THE THEORY OF LOOP AND APPLICATION TO EXPERIMENTAL DESIGN

Abstract

In the first chapter, the definition of a cyclic normalized Room square together with a brief historical background to the problem of existence and construction of Room squares was given. In the second chapter, a few principal mathematical tools such as finite field, additive group, quasigroup, Latin square etc., which are of primary importance in the construction of Room squares, were mentioned briefly and, wherever possible, further references were also indicated. A general survey of chief methods of constructing Room squares was to be found in Chapter Three, notable among them was the composition theorem by Stanton and Horton, Two existence theorems were established with the use of quasigroup for normalized Room squares and methods of construction given by the author in the fourth chapter, the approach adopted was first suggested by Bruck [11], One of the theorems was discovered independently ( see Chapter Four) , whereas the other was new. Finite fields were employed in both as elements of a pair of Room quasigroups. Together they yielded Room squares from all possible fields of odd order except order nine. Starting with a field of order nine, the author showed that it is not possible to produce a Room square of order ten by first finding four pairs with distinct absolute differences and sums from the non-zero elements of the field. in accordance with the method adopted by him and. first suggested by Mullin and Nemeth in [16 ]. The application portion was on elaboration of the work done by Johnson and Archbold in [9]. In connection with this, a brief mention was made of designs like randomized block, split-plot design and incomplete block design, It turned out that Room design could be regarded as a special type of incomplete block design. The columns of a Room square could be taken as blocks, ·the rows as one set of treatments and the symbols in the square as another set of treatments. The incomplete block design is in general unbalanced. A Room square could also be viewed as an incomplete block design with respect to one set of treatments with main plots split for another set of treatments. The analysis of variance for fixed effect model was given for both. In the last chapter, prospects for further investigation were examined. Attention was drawn to the important and fundamental relationship between Latin squares and Room squares. It is the author's opinion that specially devised Latin squares may provide the key to the construction of cyclic Room squares of all order, in particular, those the sides or which are composite numbers.