MinT - Best Known (76−51, 76, s)-Nets in Base 64

Best Known (76−51, 76, s)-Nets in Base 64

(76−51, 76, 177)-Net over F₆₄ — Constructive and digital

Digital (25, 76, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 76, 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]

(76−51, 76, 288)-Net in Base 64 — Constructive

(25, 76, 288)-net in base 64, using

t-expansion [i] based on (22, 76, 288)-net in base 64, using
- 15 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]

(76−51, 76, 408)-Net over F₆₄ — Digital

Digital (25, 76, 408)-net over F₆₄, using

net from sequence [i] based on digital (25, 407)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 25 and N(F) ≥ 408, using
  - a curve from the manYPoints database [i]

(76−51, 76, 42334)-Net in Base 64 — Upper bound on s

There is no (25, 76, 42335)-net in base 64, because

1 times m-reduction [i] would yield (25, 75, 42335)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2907 607868 081655 256549 794729 911776 499249 985354 973850 166587 262785 481015 721243 941164 751995 602165 272705 081706 854170 082969 859321 070492 392558 > 64⁷⁵ [i]