Janoš Vidali: On tight 4-designs in Hamming association schemes

Povzetek. Let C be a tight 4-design in the Hamming association scheme H(n, q). Noda (1979) showed the that one following holds:

(1) (|C|, n, q) = (16, 5, 2),
(2) (|C|, n, q) = (243, 11, 3),
(3) (|C|, n, q) = (9a^2 (9a^2 - 1)/2, (9a^2 + 1)/5, 6), where a is a positive integer such that a ≡ 0 (mod 3), a ≡ ±1 (mod 5) and a ≡ 5 (mod 16).

The existence and uniqueness for (1) or (2) had been shown. We use triple intersection numbers of association schemes to show non-existence of tight 4-designs in Hamming association schemes H(n, 6), thus ruling out case (3). This completes the classification of tight 4-designs in H(n, q).

This is joint work with Alexander Gavrilyuk and Sho Suda.