Linear Programming Bounds for Cardinality and Energy of Codes of Given Min and Max Distances

We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distance, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and fixed cardinality. In particular, we extend the framework of Levenshtein bounds for such codes.