A bi-level balanced array (B-array) T with parameters (m,N,t) and index set μ́ = (μo,μ1,···Mt) is a matrix with m rows, N columns, and with two elements (say, 0 and 1) such that in every (t × N)submatrix T*(clearly, there are (t m) such submatrices) of T, the following combinatorial condition is satisfied: every (t × 1) vector a of T*with i (0 ≤ i ≤ ť) ones in it appears the same number μi (say) times. T is called a B-array of strength t. Clearly, an orthogonal array (O-array) is a special case of a B-array. These combinatorial arrays have been extensively used in information theory, coding theory, and design of experiments. In this paper, we restrict ourselves to arrays with t = 4 and t = 6. We derive some inequalities involving m and μi, using the concept of coincidences amongst the columns of T, which are necessary conditions for B-arrays to exist. We then use these inequalities to study the existence of these arrays and to obtain the bounds on the number of rows (also called constraints) m, for a given value of μ́.