Further Results on Some Bi-level Balanced Arrays Using Coincidences

Abstract

A bi-level balanced array (B-array) $T$ with parameters $(m,N,t)$ and index set ${\underset{\_}{\mu }}^{\prime }=\{{\mu }_0,{\mu }_1,\dots ,{\mu }_t\}$ is a matrix with $m$ rows, $N$ columns, and with two elements (say, $0$ and $1$) such that in every $(t\times N)$-submatrix $T^{\ast }$ (clearly, there are $(\genfrac{}{}{0ex}{}{m}{t})$ such submatrices) of $T$, the following combinatorial condition is satisfied: every $(t\times 1)$ vector $\underset{\_}{\alpha }$ of $T^{\ast }$ with $i$ ($0\le i\le t$) 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 ${\underset{\_}{\mu }}^{\prime }$.