MinT - Best Known (42, 42+43, s)-Nets in Base 9

Best Known (42, 42+43, s)-Nets in Base 9

(42, 42+43, 104)-Net over F₉ — Constructive and digital

Digital (42, 85, 104)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 29, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]
2. digital (13, 56, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(42, 42+43, 166)-Net over F₉ — Digital

Digital (42, 85, 166)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+43, 7105)-Net in Base 9 — Upper bound on s

There is no (42, 85, 7106)-net in base 9, because

1 times m-reduction [i] would yield (42, 84, 7106)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 143 501232 607614 186419 376880 742832 159125 635800 164990 507928 442960 791741 669195 769041 > 9⁸⁴ [i]