MinT - Best Known (234−93, 234, s)-Nets in Base 3

Best Known (234−93, 234, s)-Nets in Base 3

(234−93, 234, 156)-Net over F₃ — Constructive and digital

Digital (141, 234, 156)-net over F₃, using

4 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]

(234−93, 234, 218)-Net over F₃ — Digital

Digital (141, 234, 218)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−93, 234, 2304)-Net in Base 3 — Upper bound on s

There is no (141, 234, 2305)-net in base 3, because

1 times m-reduction [i] would yield (141, 233, 2305)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1503 453461 223747 318662 075490 341757 514240 055312 016411 131860 720224 647313 517571 940692 117785 909169 307916 328157 097977 > 3²³³ [i]