MinT - Best Known (77−51, 77, s)-Nets in Base 64

Best Known (77−51, 77, s)-Nets in Base 64

(77−51, 77, 177)-Net over F₆₄ — Constructive and digital

Digital (26, 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]

(77−51, 77, 288)-Net in Base 64 — Constructive

(26, 77, 288)-net in base 64, using

t-expansion [i] based on (22, 77, 288)-net in base 64, using
- 14 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
  - base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(77−51, 77, 425)-Net over F₆₄ — Digital

Digital (26, 77, 425)-net over F₆₄, using

net from sequence [i] based on digital (26, 424)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 26 and N(F) ≥ 425, using
  - a curve from the manYPoints database [i]

(77−51, 77, 49999)-Net in Base 64 — Upper bound on s

There is no (26, 77, 50000)-net in base 64, because

1 times m-reduction [i] would yield (26, 76, 50000)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 186127 110310 079583 025885 696083 592384 123430 160207 242085 623728 327099 759092 032564 801396 118599 929341 435213 595233 513956 270115 530597 721586 132876 > 64⁷⁶ [i]