MinT - Best Known (74−41, 74, s)-Nets in Base 9

Best Known (74−41, 74, s)-Nets in Base 9

(74−41, 74, 81)-Net over F₉ — Constructive and digital

Digital (33, 74, 81)-net over F₉, using

t-expansion [i] based on digital (32, 74, 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]

(74−41, 74, 84)-Net in Base 9 — Constructive

(33, 74, 84)-net in base 9, using

1 times m-reduction [i] based on (33, 75, 84)-net in base 9, using
- base change [i] based on digital (8, 50, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(74−41, 74, 128)-Net over F₉ — Digital

Digital (33, 74, 128)-net over F₉, using

net from sequence [i] based on digital (33, 127)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
  - a curve from the manYPoints database [i]

(74−41, 74, 3144)-Net in Base 9 — Upper bound on s

There is no (33, 74, 3145)-net in base 9, because

1 times m-reduction [i] would yield (33, 73, 3145)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4582 780345 307053 194038 356282 284987 739738 171815 075677 374599 212666 294049 > 9⁷³ [i]