MinT - Best Known (232−89, 232, s)-Nets in Base 3

Best Known (232−89, 232, s)-Nets in Base 3

(232−89, 232, 156)-Net over F₃ — Constructive and digital

Digital (143, 232, 156)-net over F₃, using

10 times m-reduction [i] based on digital (143, 242, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 121, 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]

(232−89, 232, 240)-Net over F₃ — Digital

Digital (143, 232, 240)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−89, 232, 2715)-Net in Base 3 — Upper bound on s

There is no (143, 232, 2716)-net in base 3, because

1 times m-reduction [i] would yield (143, 231, 2716)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 164 169588 871885 660021 951028 839973 621212 362522 855333 795547 123199 184359 370274 964569 020960 810957 693944 269852 023105 > 3²³¹ [i]