MinT - Best Known (136−81, 136, s)-Nets in Base 9

Best Known (136−81, 136, s)-Nets in Base 9

(136−81, 136, 84)-Net over F₉ — Constructive and digital

Digital (55, 136, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 42, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [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]

(136−81, 136, 88)-Net in Base 9 — Constructive

(55, 136, 88)-net in base 9, using

2 times m-reduction [i] based on (55, 138, 88)-net in base 9, using
- base change [i] based on digital (9, 92, 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]

(136−81, 136, 182)-Net over F₉ — Digital

Digital (55, 136, 182)-net over F₉, using

t-expansion [i] based on digital (50, 136, 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]

(136−81, 136, 3250)-Net in Base 9 — Upper bound on s

There is no (55, 136, 3251)-net in base 9, because

1 times m-reduction [i] would yield (55, 135, 3251)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 664 926942 058905 973180 737028 886547 321440 989327 357559 246692 731383 797206 617404 163345 640541 313312 460297 041577 505298 464468 257351 412161 > 9¹³⁵ [i]