MinT - Best Known (12, 12+33, s)-Nets in Base 64

Best Known (12, 12+33, s)-Nets in Base 64

(12, 12+33, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 45, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 45, 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+33, 257)-Net in Base 64 — Constructive

(12, 45, 257)-net in base 64, using

3 times m-reduction [i] based on (12, 48, 257)-net in base 64, using
- base change [i] based on digital (0, 36, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(12, 12+33, 257)-Net over F₆₄ — Digital

Digital (12, 45, 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+33, 9996)-Net in Base 64 — Upper bound on s

There is no (12, 45, 9997)-net in base 64, because

1 times m-reduction [i] would yield (12, 44, 9997)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 29 657253 960455 563816 784541 197623 828164 366500 475040 393745 714606 442984 660545 189872 > 64⁴⁴ [i]