MinT - Best Known (137−103, 137, s)-Nets in Base 9

Best Known (137−103, 137, s)-Nets in Base 9

(137−103, 137, 81)-Net over F₉ — Constructive and digital

Digital (34, 137, 81)-net over F₉, using

t-expansion [i] based on digital (32, 137, 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]

(137−103, 137, 128)-Net over F₉ — Digital

Digital (34, 137, 128)-net over F₉, using

t-expansion [i] based on digital (33, 137, 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]

(137−103, 137, 838)-Net in Base 9 — Upper bound on s

There is no (34, 137, 839)-net in base 9, because

1 times m-reduction [i] would yield (34, 136, 839)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6076 815102 901109 236942 287286 773821 992235 546593 257802 341432 875753 485209 176492 038402 959246 707939 047305 360958 200823 691448 155571 602985 > 9¹³⁶ [i]