MinT - Best Known (42, 107, s)-Nets in Base 9

Best Known (42, 107, s)-Nets in Base 9

(42, 107, 81)-Net over F₉ — Constructive and digital

Digital (42, 107, 81)-net over F₉, using

t-expansion [i] based on digital (32, 107, 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, 107, 140)-Net over F₉ — Digital

Digital (42, 107, 140)-net over F₉, using

t-expansion [i] based on digital (39, 107, 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, 107, 2296)-Net in Base 9 — Upper bound on s

There is no (42, 107, 2297)-net in base 9, because

1 times m-reduction [i] would yield (42, 106, 2297)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 142249 749200 344694 912236 878242 081621 222033 359918 982576 585107 769449 316098 481246 340416 477477 268259 038465 > 9¹⁰⁶ [i]