MinT - Best Known (136−83, 136, s)-Nets in Base 9

Best Known (136−83, 136, s)-Nets in Base 9

(136−83, 136, 81)-Net over F₉ — Constructive and digital

Digital (53, 136, 81)-net over F₉, using

t-expansion [i] based on digital (32, 136, 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]

(136−83, 136, 82)-Net in Base 9 — Constructive

(53, 136, 82)-net in base 9, using

2 times m-reduction [i] based on (53, 138, 82)-net in base 9, using
- base change [i] based on digital (7, 92, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(136−83, 136, 182)-Net over F₉ — Digital

Digital (53, 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]

(136−83, 136, 2772)-Net in Base 9 — Upper bound on s

There is no (53, 136, 2773)-net in base 9, because

1 times m-reduction [i] would yield (53, 135, 2773)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 665 874181 507311 447623 236249 471314 537890 761876 933058 151434 572564 588308 585477 687039 776705 849277 081822 560937 380006 321573 783634 203625 > 9¹³⁵ [i]