MinT - Best Known (3, 64, s)-Nets in Base 81

Best Known (3, 64, s)-Nets in Base 81

(3, 64, 116)-Net over F₈₁ — Constructive and digital

Digital (3, 64, 116)-net over F₈₁, using

t-expansion [i] based on digital (2, 64, 116)-net over F₈₁, using
- net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
    - a function field by SÃ©mirat [i]

(3, 64, 136)-Net over F₈₁ — Digital

Digital (3, 64, 136)-net over F₈₁, using

net from sequence [i] based on digital (3, 135)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 3 and N(F) ≥ 136, using
  - a curve from the manYPoints database [i]

(3, 64, 1261)-Net in Base 81 — Upper bound on s

There is no (3, 64, 1262)-net in base 81, because

31 times m-reduction [i] would yield (3, 33, 1262)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 960 934719 264550 944393 380295 088367 484948 083778 718903 440754 570401 > 81³³ [i]