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

Date of publication: 14. 10. 2018

Discrete mathematics seminar

Torek, 16. 10. 2018, od 10h do 12h, Plemljev seminar, Jadranska 19

**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*) = (9*a*^2 (9*a*^2 - 1)/2, (9*a*^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.