MinT - Best Known (136−91, 136, s)-Nets in Base 9

Best Known (136−91, 136, s)-Nets in Base 9

(136−91, 136, 81)-Net over F₉ — Constructive and digital

Digital (45, 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]

(136−91, 136, 147)-Net over F₉ — Digital

Digital (45, 136, 147)-net over F₉, using

t-expansion [i] based on digital (43, 136, 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]

(136−91, 136, 1578)-Net in Base 9 — Upper bound on s

There is no (45, 136, 1579)-net in base 9, because

1 times m-reduction [i] would yield (45, 135, 1579)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 669 437090 167116 845619 445691 215179 034477 523918 987203 273847 096923 716756 707233 444572 159116 584104 834640 014557 233191 077759 595223 907193 > 9¹³⁵ [i]