MinT - Best Known (188−61, 188, s)-Nets in Base 3

Best Known (188−61, 188, s)-Nets in Base 3

(188−61, 188, 162)-Net over F₃ — Constructive and digital

Digital (127, 188, 162)-net over F₃, using

2 times m-reduction [i] based on digital (127, 190, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 95, 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]

(188−61, 188, 329)-Net over F₃ — Digital

Digital (127, 188, 329)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−61, 188, 5643)-Net in Base 3 — Upper bound on s

There is no (127, 188, 5644)-net in base 3, because

1 times m-reduction [i] would yield (127, 187, 5644)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 166964 747186 642014 583424 605299 909356 968160 238635 492146 295358 903596 066857 777693 418846 115977 > 3¹⁸⁷ [i]