Dharam V. Chopra

An array *T* with *m* rows (constraints), *N* columns (runs), and with two symbols (say, 0 and 1) is merely a matrix T of size (*m*×*N*) with elements 0 and 1. With some combinatorial structure, these arrays assume great importance in statistical design of experiments. A balanced array (B-array) of strength t with *m* (*m* ≥ *t*) rows (corresponding to 'factors' in design theory), *N* columns (corresponding to 'treatment-combinations' in design of experiments), and with two symbols (say, 0 and 1), is a matrix *T* of size (*m*×*N*) and with elements 0 and 1 such that in every (*t*×*N*) sub-matrix *T** of *T*, every (*t*×1) vector with *i* (0 ≤ *i* ≤ *t*) 1s in it occurs with the same frequency (say, *μ*_{i}). The vector *μ*$'$ = (*μ*_{0}, *μ*_{1}, …, *μ*_{t}) is called the index set of the array. If *μ*_{i} = *μ* for each *i*, then the B-array is reduced to an orthogonal array (O-array). O-arrays have been extensively used in statistics, information theory, coding theory, quality control, etc. A B-array of strength two corresponds to a balanced incomplete block design (BIBD), and various other combinatorial structures are also related to B-arrays. These arrays have also been used in constructing balanced fractional factorial designs. B-arrays with different values of t, under certain conditions, would give rise to balanced fractional factorial designs of different resolutions (e.g. B-array with *t* = 4 would give us a design of resolution V which allows us to establish all the effects up to, and including, two-factor intersections under the assumption that higher order interactions are negligible). These arrays were introduced in (1956) by I.M. Chakravarti (at the suggestion of C.R. Rao). In this paper, we derive some necessary conditions for the existence of these B-arrays, which the parameters m and mu' must satisfy. These conditions are obtained by using the positive semi-definiteness of the moment matrix and consequently, we obtain results on the maximum number of constraints of such arrays.

**Bibliography:**

1. Chakravarti, I.M. Fractional replication in symmetric fractional factorial designs and partially balanced arrays., Sankhya 17 (1956), 143-164.

2. Chopra, D.V. and Srivastava, J.N., Optimal balanced 2^{7} fractional factorial designs of resolution V, with N ≤ 42., Ann. Inst. Statist. Math. 25 (1973), 587-604.

3. Rao, C.R., Some combinatorial problems of arrays and applications to design of experiments., A Survey of Combinatorial Theory (ed. by J. Srivastava, et. al), North-Holland Publishing Co. (1973), 349-359.

**Keywords:** Balanced arrays; Orthogonal arrays; Balanced incomplete block designs; Fractional factorial designs

**Biography:** Professor Chopra's research interests are in the areas of Statistical Design of Experiments, Combinatorial Mathematics, and Graph Theory. A part of his publications include numerous papers on the construction of optimal fractional factorial designs, some of which have been shown to be optimal in the class of all fractional factorial designs. He has published extensively on some combinatorial arrays such as Balanced Arrays and Orthogonal Arrays.