MinT - Best Known (45, 45+69, s)-Nets in Base 9

Best Known (45, 45+69, s)-Nets in Base 9

(45, 45+69, 81)-Net over F₉ — Constructive and digital

Digital (45, 114, 81)-net over F₉, using

t-expansion [i] based on digital (32, 114, 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]

(45, 45+69, 82)-Net in Base 9 — Constructive

(45, 114, 82)-net in base 9, using

base change [i] based on digital (7, 76, 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]

(45, 45+69, 147)-Net over F₉ — Digital

Digital (45, 114, 147)-net over F₉, using

t-expansion [i] based on digital (43, 114, 147)-net over F₉, using
- net from sequence [i] based on digital (43, 146)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 43 and N(F) ≥ 147, using
    - a curve from the manYPoints database [i]

(45, 45+69, 2490)-Net in Base 9 — Upper bound on s

There is no (45, 114, 2491)-net in base 9, because

1 times m-reduction [i] would yield (45, 113, 2491)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 682019 211558 106337 806637 195924 863543 574451 725657 581501 196471 523120 368055 398521 664462 778449 152350 563463 682353 > 9¹¹³ [i]