MinT - Best Known (232−93, 232, s)-Nets in Base 3

Best Known (232−93, 232, s)-Nets in Base 3

(232−93, 232, 156)-Net over F₃ — Constructive and digital

Digital (139, 232, 156)-net over F₃, using

2 times m-reduction [i] based on digital (139, 234, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 117, 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]

(232−93, 232, 211)-Net over F₃ — Digital

Digital (139, 232, 211)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−93, 232, 2194)-Net in Base 3 — Upper bound on s

There is no (139, 232, 2195)-net in base 3, because

1 times m-reduction [i] would yield (139, 231, 2195)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 165 682464 768355 967200 616905 701765 546048 230699 947568 539638 919965 072389 762004 801519 121325 846841 290916 541092 729261 > 3²³¹ [i]