MinT - Best Known (135−55, 135, s)-Nets in Base 3

Best Known (135−55, 135, s)-Nets in Base 3

(135−55, 135, 80)-Net over F₃ — Constructive and digital

Digital (80, 135, 80)-net over F₃, using

9 times m-reduction [i] based on digital (80, 144, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 72, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(135−55, 135, 127)-Net over F₃ — Digital

Digital (80, 135, 127)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−55, 135, 1248)-Net in Base 3 — Upper bound on s

There is no (80, 135, 1249)-net in base 3, because

1 times m-reduction [i] would yield (80, 134, 1249)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8752 000423 581064 428058 236976 964960 143773 890603 631384 938150 105147 > 3¹³⁴ [i]