MinT - Best Known (136−45, 136, s)-Nets in Base 3

Best Known (136−45, 136, s)-Nets in Base 3

(136−45, 136, 156)-Net over F₃ — Constructive and digital

Digital (91, 136, 156)-net over F₃, using

2 times m-reduction [i] based on digital (91, 138, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 69, 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]

(136−45, 136, 236)-Net over F₃ — Digital

Digital (91, 136, 236)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−45, 136, 3812)-Net in Base 3 — Upper bound on s

There is no (91, 136, 3813)-net in base 3, because

1 times m-reduction [i] would yield (91, 135, 3813)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25903 614457 173036 127760 760675 064625 869281 140946 925088 823056 293953 > 3¹³⁵ [i]