MinT - Best Known (138, 138+81, s)-Nets in Base 3

Best Known (138, 138+81, s)-Nets in Base 3

(138, 138+81, 156)-Net over F₃ — Constructive and digital

Digital (138, 219, 156)-net over F₃, using

13 times m-reduction [i] based on digital (138, 232, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 116, 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]

(138, 138+81, 253)-Net over F₃ — Digital

Digital (138, 219, 253)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+81, 3101)-Net in Base 3 — Upper bound on s

There is no (138, 219, 3102)-net in base 3, because

1 times m-reduction [i] would yield (138, 218, 3102)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 103 251982 601510 881381 169813 010025 134599 762122 431828 206504 281351 813871 132640 156911 724403 723608 283535 144337 > 3²¹⁸ [i]