MinT - Best Known (131, 131+69, s)-Nets in Base 3

Best Known (131, 131+69, s)-Nets in Base 3

(131, 131+69, 156)-Net over F₃ — Constructive and digital

Digital (131, 200, 156)-net over F₃, using

18 times m-reduction [i] based on digital (131, 218, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 109, 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]

(131, 131+69, 286)-Net over F₃ — Digital

Digital (131, 200, 286)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+69, 4164)-Net in Base 3 — Upper bound on s

There is no (131, 200, 4165)-net in base 3, because

1 times m-reduction [i] would yield (131, 199, 4165)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 89056 495611 830488 028280 450984 872840 175231 974468 786209 527771 453581 618277 308145 477824 893718 394705 > 3¹⁹⁹ [i]