Exceptional designs in some extended quadratic residue codes

Abstract

In the present paper, we give proofs of the existence of a 3‐design in the extended ternary quadratic residue code of length 14 and the extended quaternary quadratic residue code of length 18.

Infinite series of 3‐designs in the extended quadratic residue code

“…By the transitivity argument, shells of ∼ Q q p , +1 are known to support 2-designs if ≡ p 1 (mod 4) and 3-designs if ≡ p −1 (mod 4) (see Example 2.4). In the recent years, Bonnecaze-Solé [8] found a 3-design in ∼ Q ( ) 2,42 ℓ and Ishikawa [16] also found 3-designs in ∼ Q ( ) 3,14 ℓ and ∼ Q ( ) 4,18 ℓ . Since they are the cases where ≡ p 1 (mod 4), they do not follow from the transitivity argument.…”