MinT - Best Known (212−85, 212, s)-Nets in Base 3

Best Known (212−85, 212, s)-Nets in Base 3

(212−85, 212, 148)-Net over F₃ — Constructive and digital

Digital (127, 212, 148)-net over F₃, using

8 times m-reduction [i] based on digital (127, 220, 148)-net over F₃, using
- trace code for nets [i] based on digital (17, 110, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(212−85, 212, 195)-Net over F₃ — Digital

Digital (127, 212, 195)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−85, 212, 2018)-Net in Base 3 — Upper bound on s

There is no (127, 212, 2019)-net in base 3, because

1 times m-reduction [i] would yield (127, 211, 2019)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47794 304800 764308 021688 650359 915758 319082 431116 102492 387959 531593 165401 027233 196142 425519 856315 353045 > 3²¹¹ [i]