We consider two problems related to proofs and technologies of obtaining linear programming bounds for codes (spherical and in Hamming spaces). We develop a verification technique for a conjecture concerning the optimality of the Levenshtein bounds for spherical codes and prove that the conjecture holds true under certain mild assumptions. We investigate recent conditions which are sufficient for the validity of Levenshtein-type bounds for q-ary codes with given minimum and maximum distances. We provide description of all cases for lengths n ≤ 36 and alphabet sizes 2 ≤ q ≤ 4 such that our conditions are fulfilled.
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