MinT - Best Known (81, 81+135, s)-Nets in Base 3

Best Known (81, 81+135, s)-Nets in Base 3

(81, 81+135, 56)-Net over F₃ — Constructive and digital

Digital (81, 216, 56)-net over F₃, using

net from sequence [i] based on digital (81, 55)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 55)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(81, 81+135, 84)-Net over F₃ — Digital

Digital (81, 216, 84)-net over F₃, using

t-expansion [i] based on digital (71, 216, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(81, 81+135, 375)-Net in Base 3 — Upper bound on s

There is no (81, 216, 376)-net in base 3, because

1 times m-reduction [i] would yield (81, 215, 376)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 235292 621288 711799 844949 008897 334101 792882 827385 111554 437199 456448 397633 368367 618458 417599 736825 335137 > 3²¹⁵ [i]