MinT - Best Known (139−101, 139, s)-Nets in Base 9

Best Known (139−101, 139, s)-Nets in Base 9

(139−101, 139, 81)-Net over F₉ — Constructive and digital

Digital (38, 139, 81)-net over F₉, using

t-expansion [i] based on digital (32, 139, 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]

(139−101, 139, 128)-Net over F₉ — Digital

Digital (38, 139, 128)-net over F₉, using

t-expansion [i] based on digital (33, 139, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(139−101, 139, 1017)-Net in Base 9 — Upper bound on s

There is no (38, 139, 1018)-net in base 9, because

1 times m-reduction [i] would yield (38, 138, 1018)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 500757 703258 860280 054792 999080 563060 749332 389772 162294 318356 209404 083345 520278 880299 936837 634697 441735 088557 566348 290722 387366 406241 > 9¹³⁸ [i]