MinT - Best Known (70, 70+45, s)-Nets in Base 3

Best Known (70, 70+45, s)-Nets in Base 3

(70, 70+45, 80)-Net over F₃ — Constructive and digital

Digital (70, 115, 80)-net over F₃, using

9 times m-reduction [i] based on digital (70, 124, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 62, 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]

(70, 70+45, 125)-Net over F₃ — Digital

Digital (70, 115, 125)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+45, 1321)-Net in Base 3 — Upper bound on s

There is no (70, 115, 1322)-net in base 3, because

1 times m-reduction [i] would yield (70, 114, 1322)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 466648 521632 988082 666507 327516 297210 759689 444494 481741 > 3¹¹⁴ [i]