MinT - Best Known (231−77, 231, s)-Nets in Base 3

Best Known (231−77, 231, s)-Nets in Base 3

(231−77, 231, 162)-Net over F₃ — Constructive and digital

Digital (154, 231, 162)-net over F₃, using

13 times m-reduction [i] based on digital (154, 244, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 122, 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]

(231−77, 231, 360)-Net over F₃ — Digital

Digital (154, 231, 360)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−77, 231, 5765)-Net in Base 3 — Upper bound on s

There is no (154, 231, 5766)-net in base 3, because

1 times m-reduction [i] would yield (154, 230, 5766)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55 024003 333257 418837 885310 051443 771511 714239 635380 185040 673538 886512 073746 936640 085479 868568 578961 501954 830357 > 3²³⁰ [i]