On Maximal Spherical Codes I

We investigate the possibilities for attaining two Levenshtein upper bounds for spherical codes. We find the distance distributions of all codes meeting these bounds. Then we show that the fourth Levenshtein bound can be attained in some very special cases only. We prove that no codes with an irrational maximal scalar product meet the third Levenshtein bound. So in dimensions 3 ≤ n ≤ 100 exactly seven codes are known to attain this bound and ten cases remain undecided. Moreover, the first two codes (in dimensions 5 and 6) are unique up to isometry. Nonexistence of maximal codes in all dimensions n with cardinalities between 2n+1 and 2n+[7√n] is shown as well. We prove nonexistence of several infinite families of maximal codes whose maximal scalar product is rational. The distance distributions of the only known nontrivial infinite family of maximal codes (due to Levenshtein) are given.