MinT - Best Known (75−45, 75, s)-Nets in Base 81

Best Known (75−45, 75, s)-Nets in Base 81

(75−45, 75, 370)-Net over F₈₁ — Constructive and digital

Digital (30, 75, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 75, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(75−45, 75, 551)-Net over F₈₁ — Digital

Digital (30, 75, 551)-net over F₈₁, using

net from sequence [i] based on digital (30, 550)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 30 and N(F) ≥ 551, using
  - a curve from the manYPoints database [i]

(75−45, 75, 297301)-Net in Base 81 — Upper bound on s

There is no (30, 75, 297302)-net in base 81, because

1 times m-reduction [i] would yield (30, 74, 297302)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 1690 097148 978346 345848 945287 353493 481418 271885 972131 030156 890881 256609 392948 384391 386092 239726 341982 003329 899328 461156 548184 463050 370489 606721 > 81⁷⁴ [i]