MinT - Best Known (8, 8+43, s)-Nets in Base 81

Best Known (8, 8+43, s)-Nets in Base 81

(8, 8+43, 172)-Net over F₈₁ — Constructive and digital

Digital (8, 51, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 51, 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]

(8, 8+43, 226)-Net over F₈₁ — Digital

Digital (8, 51, 226)-net over F₈₁, using

net from sequence [i] based on digital (8, 225)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 8 and N(F) ≥ 226, using
  - a curve from the manYPoints database [i]

(8, 8+43, 3786)-Net in Base 81 — Upper bound on s

There is no (8, 51, 3787)-net in base 81, because

1 times m-reduction [i] would yield (8, 50, 3787)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 266327 228531 710029 419872 597875 684873 713901 998035 291625 882273 116482 332314 897697 183351 157478 234161 > 81⁵⁰ [i]