MinT - Best Known (46−39, 46, s)-Nets in Base 81

Best Known (46−39, 46, s)-Nets in Base 81

(46−39, 46, 172)-Net over F₈₁ — Constructive and digital

Digital (7, 46, 172)-net over F₈₁, using

net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
  - a function field by SÃ©mirat [i]

(46−39, 46, 190)-Net over F₈₁ — Digital

Digital (7, 46, 190)-net over F₈₁, using

t-expansion [i] based on digital (6, 46, 190)-net over F₈₁, using
- net from sequence [i] based on digital (6, 189)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 6 and N(F) ≥ 190, using
    - a curve from the manYPoints database [i]

(46−39, 46, 3273)-Net in Base 81 — Upper bound on s

There is no (7, 46, 3274)-net in base 81, because

1 times m-reduction [i] would yield (7, 45, 3274)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 76 444401 687546 760588 288160 741466 612107 343391 909052 632400 833433 817236 065692 888791 696481 > 81⁴⁵ [i]