MinT - Best Known (129, 129+70, s)-Nets in Base 3

Best Known (129, 129+70, s)-Nets in Base 3

(129, 129+70, 156)-Net over F₃ — Constructive and digital

Digital (129, 199, 156)-net over F₃, using

15 times m-reduction [i] based on digital (129, 214, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 107, 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]

(129, 129+70, 269)-Net over F₃ — Digital

Digital (129, 199, 269)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+70, 3554)-Net in Base 3 — Upper bound on s

There is no (129, 199, 3555)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 88597 661344 185801 995905 411479 889298 354291 674257 273676 156537 365334 010512 522504 913379 382039 773163 > 3¹⁹⁹ [i]