MinT - Best Known (57, 146, s)-Nets in Base 9

Best Known (57, 146, s)-Nets in Base 9

(57, 146, 81)-Net over F₉ — Constructive and digital

Digital (57, 146, 81)-net over F₉, using

t-expansion [i] based on digital (32, 146, 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]

(57, 146, 84)-Net in Base 9 — Constructive

(57, 146, 84)-net in base 9, using

1 times m-reduction [i] based on (57, 147, 84)-net in base 9, using
- base change [i] based on digital (8, 98, 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]

(57, 146, 182)-Net over F₉ — Digital

Digital (57, 146, 182)-net over F₉, using

t-expansion [i] based on digital (50, 146, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(57, 146, 2982)-Net in Base 9 — Upper bound on s

There is no (57, 146, 2983)-net in base 9, because

1 times m-reduction [i] would yield (57, 145, 2983)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2 332581 675691 368578 936789 740235 719643 647874 694154 856090 645783 147715 817357 658272 151711 321273 840002 102744 820469 743746 697133 601814 664416 918433 > 9¹⁴⁵ [i]