MinT - Best Known (140−83, 140, s)-Nets in Base 9

Best Known (140−83, 140, s)-Nets in Base 9

(140−83, 140, 92)-Net over F₉ — Constructive and digital

Digital (57, 140, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 44, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (13, 96, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(140−83, 140, 94)-Net in Base 9 — Constructive

(57, 140, 94)-net in base 9, using

1 times m-reduction [i] based on (57, 141, 94)-net in base 9, using
- base change [i] based on digital (10, 94, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(140−83, 140, 182)-Net over F₉ — Digital

Digital (57, 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]

(140−83, 140, 3441)-Net in Base 9 — Upper bound on s

There is no (57, 140, 3442)-net in base 9, because

1 times m-reduction [i] would yield (57, 139, 3442)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 372565 429564 615868 105034 167914 377230 661140 545828 406876 712528 395762 096257 513450 168655 000066 515623 281211 414173 472293 093419 119763 577105 > 9¹³⁹ [i]