MinT - Best Known (213−127, 213, s)-Nets in Base 3

Best Known (213−127, 213, s)-Nets in Base 3

(213−127, 213, 61)-Net over F₃ — Constructive and digital

Digital (86, 213, 61)-net over F₃, using

net from sequence [i] based on digital (86, 60)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 60)-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]

(213−127, 213, 84)-Net over F₃ — Digital

Digital (86, 213, 84)-net over F₃, using

t-expansion [i] based on digital (71, 213, 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]

(213−127, 213, 430)-Net in Base 3 — Upper bound on s

There is no (86, 213, 431)-net in base 3, because

1 times m-reduction [i] would yield (86, 212, 431)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 148635 366236 428231 075977 510045 667480 576644 717100 112812 701770 547402 383618 502006 135670 063830 653738 273859 > 3²¹² [i]