MinT - Best Known (131, 131+79, s)-Nets in Base 3

Best Known (131, 131+79, s)-Nets in Base 3

(131, 131+79, 156)-Net over F₃ — Constructive and digital

Digital (131, 210, 156)-net over F₃, using

8 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+79, 233)-Net over F₃ — Digital

Digital (131, 210, 233)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+79, 2736)-Net in Base 3 — Upper bound on s

There is no (131, 210, 2737)-net in base 3, because

1 times m-reduction [i] would yield (131, 209, 2737)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5242 544940 205440 275922 222171 301394 307652 782722 961932 412124 402598 521277 028230 778100 742932 754670 900939 > 3²⁰⁹ [i]