MinT - Best Known (140−84, 140, s)-Nets in Base 9

Best Known (140−84, 140, s)-Nets in Base 9

(140−84, 140, 81)-Net over F₉ — Constructive and digital

Digital (56, 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]

(140−84, 140, 88)-Net in Base 9 — Constructive

(56, 140, 88)-net in base 9, using

1 times m-reduction [i] based on (56, 141, 88)-net in base 9, using
- base change [i] based on digital (9, 94, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(140−84, 140, 182)-Net over F₉ — Digital

Digital (56, 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]

(140−84, 140, 3104)-Net in Base 9 — Upper bound on s

There is no (56, 140, 3105)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 39 716662 170578 250636 947380 887460 988066 843158 218431 152294 380908 955271 895129 585972 641875 099751 433344 058826 485001 001883 098125 694129 893009 > 9¹⁴⁰ [i]