MinT - Best Known (21, 21+55, s)-Nets in Base 64

Best Known (21, 21+55, s)-Nets in Base 64

(21, 21+55, 177)-Net over F₆₄ — Constructive and digital

Digital (21, 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]

(21, 21+55, 288)-Net in Base 64 — Constructive

(21, 76, 288)-net in base 64, using

8 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
- base change [i] based on digital (9, 72, 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]

(21, 21+55, 342)-Net over F₆₄ — Digital

Digital (21, 76, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 76, 342)-net over F₆₄, using
- net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
    - a curve from the manYPoints database [i]

(21, 21+55, 18026)-Net in Base 64 — Upper bound on s

There is no (21, 76, 18027)-net in base 64, because

1 times m-reduction [i] would yield (21, 75, 18027)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2909 884231 364566 548795 013478 275881 799008 466464 969846 526241 365706 989288 075920 900552 580149 339838 171172 118207 903079 944569 870837 976748 599520 > 64⁷⁵ [i]