MinT - Best Known (73−41, 73, s)-Nets in Base 9

Best Known (73−41, 73, s)-Nets in Base 9

(73−41, 73, 81)-Net over F₉ — Constructive and digital

Digital (32, 73, 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]

(73−41, 73, 82)-Net in Base 9 — Constructive

(32, 73, 82)-net in base 9, using

2 times m-reduction [i] based on (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]

(73−41, 73, 120)-Net over F₉ — Digital

Digital (32, 73, 120)-net over F₉, using

t-expansion [i] based on digital (31, 73, 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]

(73−41, 73, 2816)-Net in Base 9 — Upper bound on s

There is no (32, 73, 2817)-net in base 9, because

1 times m-reduction [i] would yield (32, 72, 2817)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 510 842601 022581 721912 022409 964578 172882 724713 128715 973217 338318 555169 > 9⁷² [i]