MinT - Best Known (81−29, 81, s)-Nets in Base 3

Best Known (81−29, 81, s)-Nets in Base 3

(81−29, 81, 80)-Net over F₃ — Constructive and digital

Digital (52, 81, 80)-net over F₃, using

7 times m-reduction [i] based on digital (52, 88, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 44, 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]

(81−29, 81, 124)-Net over F₃ — Digital

Digital (52, 81, 124)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−29, 81, 1596)-Net in Base 3 — Upper bound on s

There is no (52, 81, 1597)-net in base 3, because

1 times m-reduction [i] would yield (52, 80, 1597)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 148 348629 913126 907053 642284 193436 773137 > 3⁸⁰ [i]