MinT - Best Known (142−59, 142, s)-Nets in Base 3

Best Known (142−59, 142, s)-Nets in Base 3

(142−59, 142, 80)-Net over F₃ — Constructive and digital

Digital (83, 142, 80)-net over F₃, using

8 times m-reduction [i] based on digital (83, 150, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 75, 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]

(142−59, 142, 125)-Net over F₃ — Digital

Digital (83, 142, 125)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(142−59, 142, 1190)-Net in Base 3 — Upper bound on s

There is no (83, 142, 1191)-net in base 3, because

1 times m-reduction [i] would yield (83, 141, 1191)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 19 044278 546215 729232 179250 792346 036834 657322 468203 075626 103691 797847 > 3¹⁴¹ [i]