MinT - Best Known (106−63, 106, s)-Nets in Base 9

Best Known (106−63, 106, s)-Nets in Base 9

(106−63, 106, 81)-Net over F₉ — Constructive and digital

Digital (43, 106, 81)-net over F₉, using

t-expansion [i] based on digital (32, 106, 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]

(106−63, 106, 82)-Net in Base 9 — Constructive

(43, 106, 82)-net in base 9, using

2 times m-reduction [i] based on (43, 108, 82)-net in base 9, using
- base change [i] based on digital (7, 72, 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]

(106−63, 106, 147)-Net over F₉ — Digital

Digital (43, 106, 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]

(106−63, 106, 2629)-Net in Base 9 — Upper bound on s

There is no (43, 106, 2630)-net in base 9, because

1 times m-reduction [i] would yield (43, 105, 2630)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15684 532233 144555 438123 395931 019026 887857 672724 576246 180610 606621 500572 944535 414239 951575 034870 863121 > 9¹⁰⁵ [i]