Further Results on Some Bi-level Balanced Arrays Using Coincidences

A bi-level balanced array (B-array) \( T \) with parameters \( (m, N, t) \) and index set \( \underline{\mu}’ = \{\mu_0, \mu_1, \ldots, \mu_t\} \) is a matrix with \( m \) rows, \( N \) columns, and with two elements (say, \( 0 \) and \( 1 \)) such that in every \( (t \times N) \)-submatrix \( T^* \) (clearly, there are \( \binom{m}{t} \) such submatrices) of \( T \), the following combinatorial condition is satisfied: every \( (t \times 1) \) vector \( \underline{\alpha} \) of \( T^* \) with \( i \) (\( 0 \leq i \leq t \)) ones in it appears the same number \( \mu_i \) (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 \( \mu_i \), 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 \( \underline{\mu}’ \).