MinT - Best Known (45, 45+65, s)-Nets in Base 9

Best Known (45, 45+65, s)-Nets in Base 9

(45, 45+65, 81)-Net over F₉ — Constructive and digital

Digital (45, 110, 81)-net over F₉, using

t-expansion [i] based on digital (32, 110, 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+65, 84)-Net in Base 9 — Constructive

(45, 110, 84)-net in base 9, using

1 times m-reduction [i] based on (45, 111, 84)-net in base 9, using
- base change [i] based on digital (8, 74, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(45, 45+65, 147)-Net over F₉ — Digital

Digital (45, 110, 147)-net over F₉, using

t-expansion [i] based on digital (43, 110, 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+65, 2826)-Net in Base 9 — Upper bound on s

There is no (45, 110, 2827)-net in base 9, because

1 times m-reduction [i] would yield (45, 109, 2827)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 103 866053 698709 364463 189134 392866 628963 641726 087295 580813 224978 446406 426104 646356 703564 453781 005100 057345 > 9¹⁰⁹ [i]