Incidence matrices and inequalities for combinatorial designs (Q2782294)

The authors employ incidence matrices to give alternative proofs to some inequalities involving design parameters. Additionally, they define two Latin squares of order \(2m\) on the symbols \(0,1,2,\dots,2m-1\) to be nearly orthogonal if, when superimposed, the ordered pairs \((i,i)\) do not appear and \((i,j)\), for \(i\neq j\), appears twice or once depending on whether \(|i-j|= m\) or not. Employing incidence matrices, they establish a bound on the size, \(t\), of a set of mutually nearly orthogonal Latin squares of even order \(v= 2m\), showing that \(t\leq m+1\), if \(v\equiv 2\pmod 4\), and \(t\leq m\), if \(v\equiv 0\pmod 4\). They also provide a method for constructing certain sets of mutually nearly orthogonal Latin squares.