MinT - Best Known (42, 42+91, s)-Nets in Base 9

Best Known (42, 42+91, s)-Nets in Base 9

(42, 42+91, 81)-Net over F₉ — Constructive and digital

Digital (42, 133, 81)-net over F₉, using

t-expansion [i] based on digital (32, 133, 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]

(42, 42+91, 140)-Net over F₉ — Digital

Digital (42, 133, 140)-net over F₉, using

t-expansion [i] based on digital (39, 133, 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]

(42, 42+91, 1359)-Net in Base 9 — Upper bound on s

There is no (42, 133, 1360)-net in base 9, because

1 times m-reduction [i] would yield (42, 132, 1360)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 914600 250278 015239 627448 591813 688905 919190 919776 981678 630965 032844 731705 474826 504752 591795 951907 914317 035436 287188 531328 230529 > 9¹³² [i]