MinT - Best Known (135, 135+81, s)-Nets in Base 3

Best Known (135, 135+81, s)-Nets in Base 3

(135, 135+81, 156)-Net over F₃ — Constructive and digital

Digital (135, 216, 156)-net over F₃, using

10 times m-reduction [i] based on digital (135, 226, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 113, 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]

(135, 135+81, 241)-Net over F₃ — Digital

Digital (135, 216, 241)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+81, 2853)-Net in Base 3 — Upper bound on s

There is no (135, 216, 2854)-net in base 3, because

1 times m-reduction [i] would yield (135, 215, 2854)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 847268 538283 230073 690255 953652 678804 364365 861453 322578 883024 285718 378801 140050 847048 483997 302852 458257 > 3²¹⁵ [i]