MinT - Best Known (9, 9+3, s)-Nets in Base 3

Best Known (9, 9+3, s)-Nets in Base 3

(9, 9+3, 6464)-Net over F₃ — Constructive and digital

Digital (9, 12, 6464)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹², 6464, F₃, 3, 3) (dual of [(6464, 3), 19380, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(3¹², 6464, F₃, 2, 3) (dual of [(6464, 2), 12916, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3¹², 6464, F₃, 3) (dual of [6464, 6452, 4]-code or 6464-cap in PG(11,3)), using
    - large caps in PG(u,Â 3) [i]

(9, 9+3, 21284)-Net over F₃ — Upper bound on s (digital)

There is no digital (9, 12, 21285)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹², 21285, F₃, 3) (dual of [21285, 21273, 4]-code or 21285-cap in PG(11,3)), but
- doubling the cap [i] would yield 42570-cap in AG(12,3), but
  - 6 times the recursive bound from Bierbrauer and Edel [i] would yield 113-cap in AG(6,3), but
    - bound on the size of affine caps [i]

(9, 9+3, 88572)-Net in Base 3 — Upper bound on s

There is no (9, 12, 88573)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹², 88573, S₃, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]