MinT - Best Known (75−43, 75, s)-Nets in Base 9

Best Known (75−43, 75, s)-Nets in Base 9

(75−43, 75, 81)-Net over F₉ — Constructive and digital

Digital (32, 75, 81)-net over F₉, using

net from sequence [i] based on digital (32, 80)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
  - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(75−43, 75, 82)-Net in Base 9 — Constructive

(32, 75, 82)-net in base 9, using

base change [i] based on digital (7, 50, 82)-net over F₂₇, using
- net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
    - a function field by SÃ©mirat [i]

(75−43, 75, 120)-Net over F₉ — Digital

Digital (32, 75, 120)-net over F₉, using

t-expansion [i] based on digital (31, 75, 120)-net over F₉, using
- net from sequence [i] based on digital (31, 119)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 31 and N(F) ≥ 120, using
    - a curve from the manYPoints database [i]

(75−43, 75, 2487)-Net in Base 9 — Upper bound on s

There is no (32, 75, 2488)-net in base 9, because

1 times m-reduction [i] would yield (32, 74, 2488)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 41256 851593 424562 109562 461274 691372 391318 766754 885360 862626 258199 219905 > 9⁷⁴ [i]