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

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

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

Digital (18, 65, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 65, 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, 65, 258)-Net in Base 64 — Constructive

(18, 65, 258)-net in base 64, using

3 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, 65, 281)-Net over F₆₄ — Digital

Digital (18, 65, 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, 65, 289)-Net in Base 64

(18, 65, 289)-net in base 64, using

3 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, 65, 15874)-Net in Base 64 — Upper bound on s

There is no (18, 65, 15875)-net in base 64, because

1 times m-reduction [i] would yield (18, 64, 15875)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 39 436790 843423 415893 115217 683270 760417 944613 304099 723540 748485 331188 116173 447308 564932 424132 221118 193559 016705 368576 > 64⁶⁴ [i]