MinT - Best Known (131, 131+77, s)-Nets in Base 3

Best Known (131, 131+77, s)-Nets in Base 3

(131, 131+77, 156)-Net over F₃ — Constructive and digital

Digital (131, 208, 156)-net over F₃, using

10 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+77, 241)-Net over F₃ — Digital

Digital (131, 208, 241)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+77, 2946)-Net in Base 3 — Upper bound on s

There is no (131, 208, 2947)-net in base 3, because

1 times m-reduction [i] would yield (131, 207, 2947)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 581 646211 242608 879327 871821 495160 199765 447349 839039 826236 982216 072402 108346 885872 938271 897696 738397 > 3²⁰⁷ [i]