MinT - Best Known (42, 42+58, s)-Nets in Base 9

Best Known (42, 42+58, s)-Nets in Base 9

(42, 42+58, 81)-Net over F₉ — Constructive and digital

Digital (42, 100, 81)-net over F₉, using

t-expansion [i] based on digital (32, 100, 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+58, 84)-Net in Base 9 — Constructive

(42, 100, 84)-net in base 9, using

2 times m-reduction [i] based on (42, 102, 84)-net in base 9, using
- base change [i] based on digital (8, 68, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(42, 42+58, 140)-Net over F₉ — Digital

Digital (42, 100, 140)-net over F₉, using

t-expansion [i] based on digital (39, 100, 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+58, 2830)-Net in Base 9 — Upper bound on s

There is no (42, 100, 2831)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 267516 992286 391539 073907 445867 964964 134027 648154 729785 504415 743374 106718 365186 925028 376555 157913 > 9¹⁰⁰ [i]