MinT - Best Known (12−3, 12, s)-Nets in Base 128

Best Known (12−3, 12, s)-Nets in Base 128

(12−3, 12, large)-Net over F₁₂₈ — Constructive and digital

Digital (9, 12, large)-net over F₁₂₈, using

128⁵ times duplication [i] based on digital (4, 7, large)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128⁷, large, F₁₂₈, 3, 3), using
  - appending kth column [i] based on linear OOA(128⁷, large, F₁₂₈, 2, 3), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(128⁷, large, F₁₂₈, 3) (dual of [large, large−7, 4]-code), using
      - product of projective cap with 128²-ovoid cap in AG(3,Â 128) [i] based on linear OA(128⁴, 16385, F₁₂₈, 3) (dual of [16385, 16381, 4]-code or 16385-cap in PG(3,128)), using
        ovoid in PG(3,Â 128) [i]

(12−3, 12, large)-Net in Base 128 — Upper bound on s

There is no (9, 12, large)-net in base 128, because

1 times m-reduction [i] would yield (9, 11, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]