MinT - Best Known (12, 12+65, s)-Nets in Base 64

Best Known (12, 12+65, s)-Nets in Base 64

(12, 12+65, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 77, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 77, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(12, 12+65, 257)-Net over F₆₄ — Digital

Digital (12, 77, 257)-net over F₆₄, using

net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
  - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(12, 12+65, 3939)-Net in Base 64 — Upper bound on s

There is no (12, 77, 3940)-net in base 64, because

1 times m-reduction [i] would yield (12, 76, 3940)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 186599 224386 529430 263806 141169 361108 499182 058488 325534 675347 167864 308054 940072 761624 175275 318419 432763 632049 592855 101543 012174 778514 235655 > 64⁷⁶ [i]