MinT - Best Known (238−81, 238, s)-Nets in Base 3

Best Known (238−81, 238, s)-Nets in Base 3

(238−81, 238, 162)-Net over F₃ — Constructive and digital

Digital (157, 238, 162)-net over F₃, using

12 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 125, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(238−81, 238, 347)-Net over F₃ — Digital

Digital (157, 238, 347)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−81, 238, 5253)-Net in Base 3 — Upper bound on s

There is no (157, 238, 5254)-net in base 3, because

1 times m-reduction [i] would yield (157, 237, 5254)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 119913 806005 215282 017500 994588 848770 977792 361098 525060 540669 360041 507436 069044 728873 423502 196566 754747 133025 348881 > 3²³⁷ [i]