MinT - Best Known (18, 18+65, s)-Nets in Base 64

Best Known (18, 18+65, s)-Nets in Base 64

(18, 18+65, 177)-Net over F₆₄ — Constructive and digital

Digital (18, 83, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 83, 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]

(18, 18+65, 216)-Net in Base 64 — Constructive

(18, 83, 216)-net in base 64, using

8 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
- base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(18, 18+65, 281)-Net over F₆₄ — Digital

Digital (18, 83, 281)-net over F₆₄, using

net from sequence [i] based on digital (18, 280)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 18 and N(F) ≥ 281, using
  - a curve from the manYPoints database [i]

(18, 18+65, 8611)-Net in Base 64 — Upper bound on s

There is no (18, 83, 8612)-net in base 64, because

1 times m-reduction [i] would yield (18, 82, 8612)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 12818 981948 065030 425269 478388 036783 832819 915451 408435 385605 762863 807669 437535 373511 659824 792624 088515 100316 232537 622478 067509 130222 109806 710849 743445 > 64⁸² [i]