MinT - Best Known (42−27, 42, s)-Nets in Base 64

Best Known (42−27, 42, s)-Nets in Base 64

(42−27, 42, 177)-Net over F₆₄ — Constructive and digital

Digital (15, 42, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 42, 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]

(42−27, 42, 258)-Net over F₆₄ — Digital

Digital (15, 42, 258)-net over F₆₄, using

net from sequence [i] based on digital (15, 257)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 15 and N(F) ≥ 258, using
  - a curve from the manYPoints database [i]

(42−27, 42, 288)-Net in Base 64 — Constructive

(15, 42, 288)-net in base 64, using

base change [i] based on digital (9, 36, 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]

(42−27, 42, 321)-Net in Base 64

(15, 42, 321)-net in base 64, using

10 times m-reduction [i] based on (15, 52, 321)-net in base 64, using
- base change [i] based on digital (2, 39, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(42−27, 42, 44711)-Net in Base 64 — Upper bound on s

There is no (15, 42, 44712)-net in base 64, because

1 times m-reduction [i] would yield (15, 41, 44712)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 113 095073 335185 725160 098045 494154 318183 349868 263575 254920 283824 073426 537804 > 64⁴¹ [i]