MinT - Best Known (18, 67, s)-Nets in Base 64

Best Known (18, 67, s)-Nets in Base 64

(18, 67, 177)-Net over F₆₄ — Constructive and digital

Digital (18, 67, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 67, 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, 67, 258)-Net in Base 64 — Constructive

(18, 67, 258)-net in base 64, using

1 times m-reduction [i] based on (18, 68, 258)-net in base 64, using
- base change [i] based on digital (1, 51, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(18, 67, 281)-Net over F₆₄ — Digital

Digital (18, 67, 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, 67, 289)-Net in Base 64

(18, 67, 289)-net in base 64, using

1 times m-reduction [i] based on (18, 68, 289)-net in base 64, using
- base change [i] based on digital (1, 51, 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]

(18, 67, 14409)-Net in Base 64 — Upper bound on s

There is no (18, 67, 14410)-net in base 64, because

1 times m-reduction [i] would yield (18, 66, 14410)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 161432 824014 039872 856797 417160 381178 419632 196751 805981 371202 277734 446951 279887 913562 386938 781076 593952 017388 715558 459364 > 64⁶⁶ [i]