MinT - Best Known (9, 9+3, s)-Nets in Base 64

Best Known (9, 9+3, s)-Nets in Base 64

(9, 9+3, large)-Net over F₆₄ — Constructive and digital

Digital (9, 12, large)-net over F₆₄, using

64⁵ times duplication [i] based on digital (4, 7, large)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64⁷, large, F₆₄, 3, 3), using
  - appending kth column [i] based on linear OOA(64⁷, large, F₆₄, 2, 3), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁷, large, F₆₄, 3) (dual of [large, large−7, 4]-code), using
      - product of projective cap with 64²-ovoid cap in AG(3,Â 64) [i] based on linear OA(64⁴, 4097, F₆₄, 3) (dual of [4097, 4093, 4]-code or 4097-cap in PG(3,64)), using
        ovoid in PG(3,Â 64) [i]

(9, 9+3, large)-Net in Base 64 — Upper bound on s

There is no (9, 12, large)-net in base 64, because

1 times m-reduction [i] would yield (9, 11, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]