Linear programming bounds for spherical (k,k)-designs

We derive general linear programming bounds for spherical $(k,k)$-designs. This includes lower bounds for the minimum cardinality and lower and upper bounds for minimum and maximum energy, respectively. As applications we obtain a universal bound in sense of Levenshtein for the minimum possible cardinality of a $(k,k)$ design for fixed dimension and $k$ and corresponding optimality result. We also discuss examples and possibilities for attaining the universal bound.