MinT - Best Known (127, 127+81, s)-Nets in Base 3

Best Known (127, 127+81, s)-Nets in Base 3

(127, 127+81, 156)-Net over F₃ — Constructive and digital

Digital (127, 208, 156)-net over F₃, using

2 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+81, 209)-Net over F₃ — Digital

Digital (127, 208, 209)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+81, 2282)-Net in Base 3 — Upper bound on s

There is no (127, 208, 2283)-net in base 3, because

1 times m-reduction [i] would yield (127, 207, 2283)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 583 048795 585633 522159 167121 967834 127517 457845 617862 982780 517541 515784 700793 821973 034247 472668 811825 > 3²⁰⁷ [i]