MinT - Best Known (44−33, 44, s)-Nets in Base 64

Best Known (44−33, 44, s)-Nets in Base 64

(44−33, 44, 177)-Net over F₆₄ — Constructive and digital

Digital (11, 44, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 44, 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]

(44−33, 44, 225)-Net over F₆₄ — Digital

Digital (11, 44, 225)-net over F₆₄, using

t-expansion [i] based on digital (10, 44, 225)-net over F₆₄, using
- net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(44−33, 44, 257)-Net in Base 64 — Constructive

(11, 44, 257)-net in base 64, using

base change [i] based on digital (0, 33, 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]

(44−33, 44, 7706)-Net in Base 64 — Upper bound on s

There is no (11, 44, 7707)-net in base 64, because

1 times m-reduction [i] would yield (11, 43, 7707)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 463426 665389 510885 318055 487694 480089 737155 331688 804498 936586 320747 569217 827031 > 64⁴³ [i]