MinT - Best Known (56, 137, s)-Nets in Base 9

Best Known (56, 137, s)-Nets in Base 9

(56, 137, 92)-Net over F₉ — Constructive and digital

Digital (56, 137, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 43, 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, 94, 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]

(56, 137, 94)-Net in Base 9 — Constructive

(56, 137, 94)-net in base 9, using

1 times m-reduction [i] based on (56, 138, 94)-net in base 9, using
- base change [i] based on digital (10, 92, 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]

(56, 137, 182)-Net over F₉ — Digital

Digital (56, 137, 182)-net over F₉, using

t-expansion [i] based on digital (50, 137, 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]

(56, 137, 3435)-Net in Base 9 — Upper bound on s

There is no (56, 137, 3436)-net in base 9, because

1 times m-reduction [i] would yield (56, 136, 3436)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 5987 609964 218321 745764 509484 518422 299016 797253 112250 888330 817150 908049 143579 582034 623362 317493 790304 396266 018465 296168 761831 743745 > 9¹³⁶ [i]