MinT - Best Known (239−133, 239, s)-Nets in Base 3

Best Known (239−133, 239, s)-Nets in Base 3

(239−133, 239, 73)-Net over F₃ — Constructive and digital

Digital (106, 239, 73)-net over F₃, using

net from sequence [i] based on digital (106, 72)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 72)-sequence over F₉, using
  - s-reduction based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(239−133, 239, 104)-Net over F₃ — Digital

Digital (106, 239, 104)-net over F₃, using

t-expansion [i] based on digital (102, 239, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(239−133, 239, 604)-Net in Base 3 — Upper bound on s

There is no (106, 239, 605)-net in base 3, because

1 times m-reduction [i] would yield (106, 238, 605)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 367827 146688 376934 760333 879992 897974 127035 946605 895358 814006 175185 647377 357082 302909 498128 336322 705070 212635 580097 > 3²³⁸ [i]