MinT - Best Known (42, 42+89, s)-Nets in Base 9

Best Known (42, 42+89, s)-Nets in Base 9

(42, 42+89, 81)-Net over F₉ — Constructive and digital

Digital (42, 131, 81)-net over F₉, using

t-expansion [i] based on digital (32, 131, 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+89, 140)-Net over F₉ — Digital

Digital (42, 131, 140)-net over F₉, using

t-expansion [i] based on digital (39, 131, 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+89, 1396)-Net in Base 9 — Upper bound on s

There is no (42, 131, 1397)-net in base 9, because

1 times m-reduction [i] would yield (42, 130, 1397)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11582 466089 091076 871600 397032 110177 212263 976995 922397 564934 548697 023910 983257 301385 952538 176835 068809 897398 540900 532794 456161 > 9¹³⁰ [i]