MinT - Best Known (198−111, 198, s)-Nets in Base 3

Best Known (198−111, 198, s)-Nets in Base 3

(198−111, 198, 62)-Net over F₃ — Constructive and digital

Digital (87, 198, 62)-net over F₃, using

net from sequence [i] based on digital (87, 61)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 61)-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]

(198−111, 198, 84)-Net over F₃ — Digital

Digital (87, 198, 84)-net over F₃, using

t-expansion [i] based on digital (71, 198, 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]

(198−111, 198, 493)-Net in Base 3 — Upper bound on s

There is no (87, 198, 494)-net in base 3, because

1 times m-reduction [i] would yield (87, 197, 494)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10325 552802 924203 046013 314121 782027 934344 937367 065646 729559 684392 262856 155864 314964 502247 456409 > 3¹⁹⁷ [i]