MinT - Best Known (84, 84+57, s)-Nets in Base 3

Best Known (84, 84+57, s)-Nets in Base 3

(84, 84+57, 128)-Net over F₃ — Constructive and digital

Digital (84, 141, 128)-net over F₃, using

1 times m-reduction [i] based on digital (84, 142, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 71, 64)-net over F₉, using
  - net from sequence [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]

(84, 84+57, 135)-Net over F₃ — Digital

Digital (84, 141, 135)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(84, 84+57, 1345)-Net in Base 3 — Upper bound on s

There is no (84, 141, 1346)-net in base 3, because

1 times m-reduction [i] would yield (84, 140, 1346)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 337034 230447 440510 186489 104638 121051 246028 301392 454245 027179 578393 > 3¹⁴⁰ [i]