MinT - Best Known (212−117, 212, s)-Nets in Base 3

Best Known (212−117, 212, s)-Nets in Base 3

(212−117, 212, 64)-Net over F₃ — Constructive and digital

Digital (95, 212, 64)-net over F₃, using

t-expansion [i] based on digital (89, 212, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(212−117, 212, 96)-Net over F₃ — Digital

Digital (95, 212, 96)-net over F₃, using

t-expansion [i] based on digital (89, 212, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(212−117, 212, 555)-Net in Base 3 — Upper bound on s

There is no (95, 212, 556)-net in base 3, because

1 times m-reduction [i] would yield (95, 211, 556)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 49390 437287 231387 661447 589087 464755 827136 038458 799097 725054 714597 479392 788502 474427 773157 383580 534825 > 3²¹¹ [i]