MinT - Best Known (38, 38+51, s)-Nets in Base 9

Best Known (38, 38+51, s)-Nets in Base 9

(38, 38+51, 81)-Net over F₉ — Constructive and digital

Digital (38, 89, 81)-net over F₉, using

t-expansion [i] based on digital (32, 89, 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]

(38, 38+51, 84)-Net in Base 9 — Constructive

(38, 89, 84)-net in base 9, using

1 times m-reduction [i] based on (38, 90, 84)-net in base 9, using
- base change [i] based on digital (8, 60, 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]

(38, 38+51, 128)-Net over F₉ — Digital

Digital (38, 89, 128)-net over F₉, using

t-expansion [i] based on digital (33, 89, 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]

(38, 38+51, 2892)-Net in Base 9 — Upper bound on s

There is no (38, 89, 2893)-net in base 9, because

1 times m-reduction [i] would yield (38, 88, 2893)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 948076 496521 049979 268353 223526 187872 750983 649195 957782 533727 961215 798201 147881 710889 > 9⁸⁸ [i]