We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval $[ell,s]$ of $[-1,1)$. An intricate relationship between Levenshtein-type upper bounds on cardinality of codes with inner products in $[ell,s]$ and lower bounds on the potential energy (for absolutely monotone interactions) for codes with inner products in $[ell,1)$ (when the cardinality of the code is kept fixed) is revealed and explained. Thereby, we obtain a new extension of Levenshtein bounds for such codes. The universality of our bounds is exhibited by a unified derivation and their validity for a wide range of codes and potential functions.
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