MinT - Best Known (42, 42+56, s)-Nets in Base 9

Best Known (42, 42+56, s)-Nets in Base 9

(42, 42+56, 81)-Net over F₉ — Constructive and digital

Digital (42, 98, 81)-net over F₉, using

t-expansion [i] based on digital (32, 98, 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+56, 88)-Net in Base 9 — Constructive

(42, 98, 88)-net in base 9, using

1 times m-reduction [i] based on (42, 99, 88)-net in base 9, using
- base change [i] based on digital (9, 66, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(42, 42+56, 140)-Net over F₉ — Digital

Digital (42, 98, 140)-net over F₉, using

t-expansion [i] based on digital (39, 98, 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+56, 3071)-Net in Base 9 — Upper bound on s

There is no (42, 98, 3072)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3289 812610 963447 046513 203357 087774 887628 160211 244197 495127 323871 616550 947825 879739 880141 062145 > 9⁹⁸ [i]