Arrays of strength s on two symbols (Q800364)

A matrix T with two elements (say) 0 and 1 is called an array of strength s if in every s-rowed submatrix of T the number of columns of T with exactly i non-zero elements depends only on i for each \(i\leq s\). Furthermore T is called a balanced array of strength s if (i) T is an array of strength s, and (ii) for every \(i\leq s\), the columnns with i non-zero entries in T are evenly distributed among the \(\left( \begin{matrix} s\\ i\end{matrix} \right)\) possible types. A balanced array T with m rows is said to be perfectly balanced if it is balanced for each \(s\leq m\), i.e. every possible type of vector of a given weight appears the same number of times. The vector \(\sigma =(\sigma_ 0,\sigma_ 1,\sigma_ 2,...,\sigma_ s)\) is called the weight of an array T of strength s, where \(\sigma_ i\) denotes the number of columns with i non-zero elements among each s-rowed submatrix. If T is also balanced then \(\sigma_ i=\left( \begin{matrix} s\\ i\end{matrix} \right)\mu_ i\), and the vector \(\mu =(\mu_ 0,\mu_ 1,...,\mu_ s)\) is called the weight of T. In this paper necessary and sufficient conditions are given on \(\sigma\),s,t and on \(\mu\), s, t for an array with \((s+t)\) rows to have strength s and weight \(\sigma\), or to be balanced and have strength s and weight \(\mu\). Special cases of \(t=1\), 2 are provided in greater detail, and some useful methods of construction are also given.