MinT - Best Known (137, 137+82, s)-Nets in Base 3

Best Known (137, 137+82, s)-Nets in Base 3

(137, 137+82, 156)-Net over F₃ — Constructive and digital

Digital (137, 219, 156)-net over F₃, using

11 times m-reduction [i] based on digital (137, 230, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 115, 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]

(137, 137+82, 245)-Net over F₃ — Digital

Digital (137, 219, 245)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+82, 2813)-Net in Base 3 — Upper bound on s

There is no (137, 219, 2814)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 311 205888 579756 575122 961902 460989 252526 423392 195560 836822 780562 922071 297769 670680 420576 931125 961268 453933 > 3²¹⁹ [i]