MinT - Best Known (53, 53+81, s)-Nets in Base 9

Best Known (53, 53+81, s)-Nets in Base 9

(53, 53+81, 81)-Net over F₉ — Constructive and digital

Digital (53, 134, 81)-net over F₉, using

t-expansion [i] based on digital (32, 134, 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]

(53, 53+81, 84)-Net in Base 9 — Constructive

(53, 134, 84)-net in base 9, using

1 times m-reduction [i] based on (53, 135, 84)-net in base 9, using
- base change [i] based on digital (8, 90, 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]

(53, 53+81, 182)-Net over F₉ — Digital

Digital (53, 134, 182)-net over F₉, using

t-expansion [i] based on digital (50, 134, 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]

(53, 53+81, 2910)-Net in Base 9 — Upper bound on s

There is no (53, 134, 2911)-net in base 9, because

1 times m-reduction [i] would yield (53, 133, 2911)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8 296282 571562 238062 221422 786755 501207 930849 977751 024782 950874 249071 101464 343035 473172 025577 111483 561148 922503 530653 401273 411265 > 9¹³³ [i]