MinT - Best Known (136−95, 136, s)-Nets in Base 9

Best Known (136−95, 136, s)-Nets in Base 9

(136−95, 136, 81)-Net over F₉ — Constructive and digital

Digital (41, 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]

(136−95, 136, 140)-Net over F₉ — Digital

Digital (41, 136, 140)-net over F₉, using

t-expansion [i] based on digital (39, 136, 140)-net over F₉, using
- net from sequence [i] based on digital (39, 139)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
    - a curve from the manYPoints database [i]

(136−95, 136, 1235)-Net in Base 9 — Upper bound on s

There is no (41, 136, 1236)-net in base 9, because

1 times m-reduction [i] would yield (41, 135, 1236)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 669 443234 807511 358977 313716 770118 572020 569248 529872 122598 567336 697650 294895 241538 787442 194289 067335 000164 538893 635391 333171 546081 > 9¹³⁵ [i]