Summary: For a rational \(\alpha\in (0,1)\), let \({\mathcal A}_{n\times m,\alpha}\) be the set of binary \(n\times m\) arrays in which each row has Hamming weight \(\alpha m\) and each column has Hamming weight \(\alpha n\), where \(\alpha m\) and \(\alpha n\) are integers. (The special case of two-dimensional balanced arrays corresponds to \(\alpha= 1/2\) and even values for \(n\) and \(m\).) The redundancy of \({\mathcal A}_{n\times m,\alpha}\) is defined by \[ \rho_{n\times m,\alpha}= nmH(\alpha)- \log_2|{\mathcal A}_{n\times m,\alpha}|, \] where \(H(x)=- x\log_2 x-(1- x)\log_2(1- x)\). Bounds on \(\rho_{n\times m,\alpha}\) are obtained in terms of the redundancies of the sets \({\mathcal A}_{\ell,\alpha}\) of all binary \(\ell\)-vectors with Hamming weight \(\alpha\ell\), \(\ell\in \{n,m\}\). Specifically, it is shown that \[ \rho_{n\times m,\alpha}\leq n\rho_{m,\alpha}+ m\rho_{n,\alpha}, \] where \(\rho_{\ell,\alpha}= \ell H(\alpha)- \log_2|{\mathcal A}_{\ell,\alpha}|\) and that this bound is tight up to an additive term \(O(n+ \log m)\). A polynomial-time coding algorithm is presented that maps unconstrained input sequences into \({\mathcal A}_{n\times m,\alpha}\) at a rate \(H(\alpha)-(\rho_{m,\alpha}/m)\).