On the coexistence of conference matrices and near resolvable 2-(2k+1, k, k-1) designs

We show that a near resolvable 2-(2k + 1, k, k − 1) design exists if and only if a conference matrix of order 2k+2 does. A known result on conference matrices then allows us to conclude that a near resolvable 2-(2k + 1, k, k − 1) design with even k can only exist if 2k + 1 is the sum of two squares. In particular, neither a near resolvable 2-(21, 10, 9) design nor a near resolvable 2-(33, 16, 15) design exists. For k ≤ 14, we also enumerate the near resolvable 2-(2k + 1, k, k − 1) designs and the corresponding conference matrices.