New infinite families of 3-designs from preparata codes over Z 4

Abstract

We consider t-designs constructed from codewords in the Preparata code P m over Z 4. A new approach is given to prove that the support (size 5) of minimum Lee weight codewords form a simple 3-design for any odd integer m ⩾ 3. We also show that the support of codewords with support size 6 form four new families of simple 3-designs, with parameters (2 m , 6, 2 m − 8), (2 m , 6, 5 · (2 m−1 − 4)), (2 m , 6,20 ·(2 m−1 − 4)/3) and (2 m , 6, 18 · (2 m−1 − 4)), for any odd integer m ⩾ 5. Codewords with support size 7 are also investigated by computer search.