MinT - Best Known (230−89, 230, s)-Nets in Base 3

Best Known (230−89, 230, s)-Nets in Base 3

(230−89, 230, 156)-Net over F₃ — Constructive and digital

Digital (141, 230, 156)-net over F₃, using

8 times m-reduction [i] based on digital (141, 238, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 119, 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]

(230−89, 230, 232)-Net over F₃ — Digital

Digital (141, 230, 232)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−89, 230, 2581)-Net in Base 3 — Upper bound on s

There is no (141, 230, 2582)-net in base 3, because

1 times m-reduction [i] would yield (141, 229, 2582)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 360284 305443 125934 247745 585375 795391 053722 455759 094402 265473 429807 755230 407281 546421 310629 000761 867133 866393 > 3²²⁹ [i]