MinT - Best Known (129, 129+81, s)-Nets in Base 3

Best Known (129, 129+81, s)-Nets in Base 3

(129, 129+81, 156)-Net over F₃ — Constructive and digital

Digital (129, 210, 156)-net over F₃, using

4 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+81, 216)-Net over F₃ — Digital

Digital (129, 210, 216)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+81, 2413)-Net in Base 3 — Upper bound on s

There is no (129, 210, 2414)-net in base 3, because

1 times m-reduction [i] would yield (129, 209, 2414)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5237 615675 551941 468076 943817 264717 200566 798673 569116 075669 223450 648965 401727 539655 122536 382576 161425 > 3²⁰⁹ [i]