MinT - Best Known (42, 42+97, s)-Nets in Base 9

Best Known (42, 42+97, s)-Nets in Base 9

(42, 42+97, 81)-Net over F₉ — Constructive and digital

Digital (42, 139, 81)-net over F₉, using

t-expansion [i] based on digital (32, 139, 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+97, 140)-Net over F₉ — Digital

Digital (42, 139, 140)-net over F₉, using

t-expansion [i] based on digital (39, 139, 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+97, 1268)-Net in Base 9 — Upper bound on s

There is no (42, 139, 1269)-net in base 9, because

1 times m-reduction [i] would yield (42, 138, 1269)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 497169 541483 498193 743180 883428 522645 972027 049520 673482 000146 868284 815731 484372 808553 054670 997876 946370 247537 994424 443790 461371 671937 > 9¹³⁸ [i]