MinT - Best Known (52, 52+81, s)-Nets in Base 9

Best Known (52, 52+81, s)-Nets in Base 9

(52, 52+81, 81)-Net over F₉ — Constructive and digital

Digital (52, 133, 81)-net over F₉, using

t-expansion [i] based on digital (32, 133, 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]

(52, 52+81, 82)-Net in Base 9 — Constructive

(52, 133, 82)-net in base 9, using

2 times m-reduction [i] based on (52, 135, 82)-net in base 9, using
- base change [i] based on digital (7, 90, 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]

(52, 52+81, 182)-Net over F₉ — Digital

Digital (52, 133, 182)-net over F₉, using

t-expansion [i] based on digital (50, 133, 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]

(52, 52+81, 2753)-Net in Base 9 — Upper bound on s

There is no (52, 133, 2754)-net in base 9, because

1 times m-reduction [i] would yield (52, 132, 2754)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 920396 795577 408156 464518 920727 883700 171053 782356 174779 278328 440834 116353 062483 289773 335786 753428 476641 518524 041945 207423 269761 > 9¹³² [i]