MinT - Best Known (9, 12, s)-Nets in Base 81

Best Known (9, 12, s)-Nets in Base 81

(9, 12, large)-Net over F₈₁ — Constructive and digital

Digital (9, 12, large)-net over F₈₁, using

81⁵ times duplication [i] based on digital (4, 7, large)-net over F₈₁, using
- net defined by OOA [i] based on linear OOA(81⁷, large, F₈₁, 3, 3), using
  - appending kth column [i] based on linear OOA(81⁷, large, F₈₁, 2, 3), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁷, large, F₈₁, 3) (dual of [large, large−7, 4]-code), using
      - product of projective cap with 81²-ovoid cap in AG(3,Â 81) [i] based on linear OA(81⁴, 6562, F₈₁, 3) (dual of [6562, 6558, 4]-code or 6562-cap in PG(3,81)), using
        ovoid in PG(3,Â 81) [i]

(9, 12, large)-Net in Base 81 — Upper bound on s

There is no (9, 12, large)-net in base 81, because

1 times m-reduction [i] would yield (9, 11, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]