MinT - Best Known (222−139, 222, s)-Nets in Base 3

Best Known (222−139, 222, s)-Nets in Base 3

(222−139, 222, 58)-Net over F₃ — Constructive and digital

Digital (83, 222, 58)-net over F₃, using

net from sequence [i] based on digital (83, 57)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 57)-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]

(222−139, 222, 84)-Net over F₃ — Digital

Digital (83, 222, 84)-net over F₃, using

t-expansion [i] based on digital (71, 222, 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]

(222−139, 222, 383)-Net in Base 3 — Upper bound on s

There is no (83, 222, 384)-net in base 3, because

1 times m-reduction [i] would yield (83, 221, 384)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3218 631654 148158 071185 224663 690223 807290 321202 727659 608059 255527 919128 290694 192377 990394 943105 516680 061697 > 3²²¹ [i]