MinT - Best Known (131, 131+75, s)-Nets in Base 3

Best Known (131, 131+75, s)-Nets in Base 3

(131, 131+75, 156)-Net over F₃ — Constructive and digital

Digital (131, 206, 156)-net over F₃, using

12 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+75, 251)-Net over F₃ — Digital

Digital (131, 206, 251)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+75, 3187)-Net in Base 3 — Upper bound on s

There is no (131, 206, 3188)-net in base 3, because

1 times m-reduction [i] would yield (131, 205, 3188)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 64 807680 036786 751869 434276 501862 267400 590980 146274 951113 072421 712370 210806 435989 043825 532895 200713 > 3²⁰⁵ [i]