MinT - Best Known (140−85, 140, s)-Nets in Base 9

Best Known (140−85, 140, s)-Nets in Base 9

(140−85, 140, 81)-Net over F₉ — Constructive and digital

Digital (55, 140, 81)-net over F₉, using

t-expansion [i] based on digital (32, 140, 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]

(140−85, 140, 84)-Net in Base 9 — Constructive

(55, 140, 84)-net in base 9, using

1 times m-reduction [i] based on (55, 141, 84)-net in base 9, using
- base change [i] based on digital (8, 94, 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]

(140−85, 140, 182)-Net over F₉ — Digital

Digital (55, 140, 182)-net over F₉, using

t-expansion [i] based on digital (50, 140, 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]

(140−85, 140, 2944)-Net in Base 9 — Upper bound on s

There is no (55, 140, 2945)-net in base 9, because

1 times m-reduction [i] would yield (55, 139, 2945)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 386084 468777 422894 467019 889523 189222 048594 472299 019076 798712 795450 171503 684363 781613 763880 526355 693072 823051 988373 550103 879130 828945 > 9¹³⁹ [i]