We use linear-programming techniques to describe points of absolute minimum over the unit sphere \(S^{d}\) in \(\mathbb R^{d+1}\) of the total potential of a spherical \((2m-1)\) -design \(\omega _N\subset S^{d}\) contained in the union of some m parallel hyperplanes (such designs attain the Fazekas-Levenshtein bound for covering). The interaction between points is described by the kernel \(K(\mathbf {x},\mathbf {y})=f(\left |\mathbf {x}-\mathbf {y}\right |^2)\) , where \(\left |\ \!\cdot \ \!\right |\) is the Euclidean norm in \(\mathbb R^{d+1}\) . If the potential function f has a strictly convex derivative \(f^{(2m-2)}\) , then points of minimum appear to be independent of f and are those and only those which form exactly m distinct dot products with points of \(\omega _N\) . Using this result, we find sets of universal minima of six new higher-dimensional spherical designs. It is interesting that these designs form three pairs, where in every pair, the universal minima of each design are exactly at points of the other design.