MinT - Best Known (21, 21+51, s)-Nets in Base 64

Best Known (21, 21+51, s)-Nets in Base 64

(21, 21+51, 177)-Net over F₆₄ — Constructive and digital

Digital (21, 72, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 72, 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+51, 288)-Net in Base 64 — Constructive

(21, 72, 288)-net in base 64, using

12 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+51, 342)-Net over F₆₄ — Digital

Digital (21, 72, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 72, 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+51, 21756)-Net in Base 64 — Upper bound on s

There is no (21, 72, 21757)-net in base 64, because

1 times m-reduction [i] would yield (21, 71, 21757)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 173 391279 490502 724434 517385 771464 959833 300252 023517 236355 164006 781743 055067 080616 126756 508936 733711 689605 014292 069785 743015 022924 > 64⁷¹ [i]