MinT - Best Known (174−59, 174, s)-Nets in Base 3

Best Known (174−59, 174, s)-Nets in Base 3

(174−59, 174, 156)-Net over F₃ — Constructive and digital

Digital (115, 174, 156)-net over F₃, using

12 times m-reduction [i] based on digital (115, 186, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 93, 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]

(174−59, 174, 268)-Net over F₃ — Digital

Digital (115, 174, 268)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(174−59, 174, 4067)-Net in Base 3 — Upper bound on s

There is no (115, 174, 4068)-net in base 3, because

1 times m-reduction [i] would yield (115, 173, 4068)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34918 297630 884971 895301 702919 237655 341217 159961 237484 137564 133601 087967 100503 658793 > 3¹⁷³ [i]