MinT - Best Known (38−27, 38, s)-Nets in Base 64

Best Known (38−27, 38, s)-Nets in Base 64

(38−27, 38, 177)-Net over F₆₄ — Constructive and digital

Digital (11, 38, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 38, 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]

(38−27, 38, 225)-Net over F₆₄ — Digital

Digital (11, 38, 225)-net over F₆₄, using

t-expansion [i] based on digital (10, 38, 225)-net over F₆₄, using
- net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(38−27, 38, 258)-Net in Base 64 — Constructive

(11, 38, 258)-net in base 64, using

2 times m-reduction [i] based on (11, 40, 258)-net in base 64, using
- base change [i] based on digital (1, 30, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(38−27, 38, 289)-Net in Base 64

(11, 38, 289)-net in base 64, using

2 times m-reduction [i] based on (11, 40, 289)-net in base 64, using
- base change [i] based on digital (1, 30, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(38−27, 38, 12431)-Net in Base 64 — Upper bound on s

There is no (11, 38, 12432)-net in base 64, because

1 times m-reduction [i] would yield (11, 37, 12432)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 6 745239 462521 213198 887228 983872 281097 474228 988117 885867 885763 395655 > 64³⁷ [i]