MinT - Best Known (241−81, 241, s)-Nets in Base 3

Best Known (241−81, 241, s)-Nets in Base 3

(241−81, 241, 162)-Net over F₃ — Constructive and digital

Digital (160, 241, 162)-net over F₃, using

t-expansion [i] based on digital (157, 241, 162)-net over F₃, using
- 9 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
  - trace code for nets [i] based on digital (32, 125, 81)-net over F₉, using
    - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
        F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(241−81, 241, 365)-Net over F₃ — Digital

Digital (160, 241, 365)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−81, 241, 5708)-Net in Base 3 — Upper bound on s

There is no (160, 241, 5709)-net in base 3, because

1 times m-reduction [i] would yield (160, 240, 5709)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 246347 193486 533019 348538 698930 516614 825034 708275 438967 409528 312725 938306 868846 772931 225611 306825 620531 235634 510049 > 3²⁴⁰ [i]