MinT - Best Known (127, 127+65, s)-Nets in Base 3

Best Known (127, 127+65, s)-Nets in Base 3

(127, 127+65, 156)-Net over F₃ — Constructive and digital

Digital (127, 192, 156)-net over F₃, using

18 times m-reduction [i] based on digital (127, 210, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 105, 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, 127+65, 293)-Net over F₃ — Digital

Digital (127, 192, 293)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+65, 4473)-Net in Base 3 — Upper bound on s

There is no (127, 192, 4474)-net in base 3, because

1 times m-reduction [i] would yield (127, 191, 4474)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13 560670 937674 899112 564534 636535 783509 206467 238226 800579 693077 653825 845495 725781 393869 824833 > 3¹⁹¹ [i]