MinT - Best Known (127−41, 127, s)-Nets in Base 3

Best Known (127−41, 127, s)-Nets in Base 3

(127−41, 127, 156)-Net over F₃ — Constructive and digital

Digital (86, 127, 156)-net over F₃, using

1 times m-reduction [i] based on digital (86, 128, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 64, 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]

(127−41, 127, 240)-Net over F₃ — Digital

Digital (86, 127, 240)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−41, 127, 4189)-Net in Base 3 — Upper bound on s

There is no (86, 127, 4190)-net in base 3, because

1 times m-reduction [i] would yield (86, 126, 4190)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 315248 366991 515096 063842 128700 324123 303771 338814 811091 988649 > 3¹²⁶ [i]