MinT - Best Known (135−101, 135, s)-Nets in Base 9

Best Known (135−101, 135, s)-Nets in Base 9

(135−101, 135, 81)-Net over F₉ — Constructive and digital

Digital (34, 135, 81)-net over F₉, using

t-expansion [i] based on digital (32, 135, 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]

(135−101, 135, 128)-Net over F₉ — Digital

Digital (34, 135, 128)-net over F₉, using

t-expansion [i] based on digital (33, 135, 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]

(135−101, 135, 848)-Net in Base 9 — Upper bound on s

There is no (34, 135, 849)-net in base 9, because

1 times m-reduction [i] would yield (34, 134, 849)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 76 161627 595456 468075 271091 526512 514133 234264 714685 364915 071474 317976 442562 159578 793203 390608 630849 811131 198071 435829 795600 273105 > 9¹³⁴ [i]