Mutually orthogonal rectangular gerechte designs

Let q   be a prime power and r,s,m,nr,s,m,n positive integers. We construct families of mutually orthogonal gerechte designs of order qr+sqr+s with rectangular regions of size qr×qsqr×qs. This leads to a lower bound on the size of a family of mutually orthogonal gerechte designs of order mn   with rectangular regions of size m×nm×n. The construction is linear-algebraic; surrounding theory employs companion matrices and Toeplitz matrices over finite fields.