MinT - Best Known (42, 42+81, s)-Nets in Base 9

Best Known (42, 42+81, s)-Nets in Base 9

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

Digital (42, 123, 81)-net over F₉, using

t-expansion [i] based on digital (32, 123, 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+81, 140)-Net over F₉ — Digital

Digital (42, 123, 140)-net over F₉, using

t-expansion [i] based on digital (39, 123, 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+81, 1579)-Net in Base 9 — Upper bound on s

There is no (42, 123, 1580)-net in base 9, because

1 times m-reduction [i] would yield (42, 122, 1580)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 265 681078 969631 783928 472905 933185 902440 698780 075399 747354 817426 568326 350747 020352 074120 996066 165916 158514 326411 243777 > 9¹²² [i]