MinT - Best Known (138−43, 138, s)-Nets in Base 3

Best Known (138−43, 138, s)-Nets in Base 3

(138−43, 138, 156)-Net over F₃ — Constructive and digital

Digital (95, 138, 156)-net over F₃, using

8 times m-reduction [i] based on digital (95, 146, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 73, 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]

(138−43, 138, 287)-Net over F₃ — Digital

Digital (95, 138, 287)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−43, 138, 5604)-Net in Base 3 — Upper bound on s

There is no (95, 138, 5605)-net in base 3, because

1 times m-reduction [i] would yield (95, 137, 5605)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 232605 047445 524451 904007 189455 691215 531375 411543 491894 725800 212891 > 3¹³⁷ [i]