MinT - Best Known (45, 45+67, s)-Nets in Base 9

Best Known (45, 45+67, s)-Nets in Base 9

(45, 45+67, 81)-Net over F₉ — Constructive and digital

Digital (45, 112, 81)-net over F₉, using

t-expansion [i] based on digital (32, 112, 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+67, 82)-Net in Base 9 — Constructive

(45, 112, 82)-net in base 9, using

2 times m-reduction [i] based on (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+67, 147)-Net over F₉ — Digital

Digital (45, 112, 147)-net over F₉, using

t-expansion [i] based on digital (43, 112, 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+67, 2646)-Net in Base 9 — Upper bound on s

There is no (45, 112, 2647)-net in base 9, because

1 times m-reduction [i] would yield (45, 111, 2647)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8365 172279 311135 521749 212114 999260 762815 849873 505965 553846 952406 010589 596373 631127 892590 041223 924240 808377 > 9¹¹¹ [i]