MinT - Best Known (42, 42+61, s)-Nets in Base 9

Best Known (42, 42+61, s)-Nets in Base 9

(42, 42+61, 81)-Net over F₉ — Constructive and digital

Digital (42, 103, 81)-net over F₉, using

t-expansion [i] based on digital (32, 103, 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+61, 82)-Net in Base 9 — Constructive

(42, 103, 82)-net in base 9, using

2 times m-reduction [i] based on (42, 105, 82)-net in base 9, using
- base change [i] based on digital (7, 70, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(42, 42+61, 140)-Net over F₉ — Digital

Digital (42, 103, 140)-net over F₉, using

t-expansion [i] based on digital (39, 103, 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+61, 2624)-Net in Base 9 — Upper bound on s

There is no (42, 103, 2625)-net in base 9, because

1 times m-reduction [i] would yield (42, 102, 2625)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 21 527982 589802 621251 721771 716784 780908 032196 119615 864317 128049 182200 821848 160360 107190 335629 662641 > 9¹⁰² [i]