MinT - Best Known (135, 135+89, s)-Nets in Base 3

Best Known (135, 135+89, s)-Nets in Base 3

(135, 135+89, 156)-Net over F₃ — Constructive and digital

Digital (135, 224, 156)-net over F₃, using

2 times m-reduction [i] based on digital (135, 226, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 113, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(135, 135+89, 210)-Net over F₃ — Digital

Digital (135, 224, 210)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+89, 2216)-Net in Base 3 — Upper bound on s

There is no (135, 224, 2217)-net in base 3, because

1 times m-reduction [i] would yield (135, 223, 2217)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25281 089045 116953 091426 239177 045061 937002 940288 839439 246828 652040 209923 536931 079070 563911 203352 735379 535665 > 3²²³ [i]