MinT - Best Known (140, 140+69, s)-Nets in Base 3

Best Known (140, 140+69, s)-Nets in Base 3

(140, 140+69, 162)-Net over F₃ — Constructive and digital

Digital (140, 209, 162)-net over F₃, using

7 times m-reduction [i] based on digital (140, 216, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 108, 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]

(140, 140+69, 341)-Net over F₃ — Digital

Digital (140, 209, 341)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(140, 140+69, 5581)-Net in Base 3 — Upper bound on s

There is no (140, 209, 5582)-net in base 3, because

1 times m-reduction [i] would yield (140, 208, 5582)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1753 247008 575597 105626 186590 611143 729520 017121 115918 310503 954858 193770 651015 838547 619667 155544 657565 > 3²⁰⁸ [i]