MinT - Best Known (231−79, 231, s)-Nets in Base 3

Best Known (231−79, 231, s)-Nets in Base 3

(231−79, 231, 162)-Net over F₃ — Constructive and digital

Digital (152, 231, 162)-net over F₃, using

9 times m-reduction [i] based on digital (152, 240, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 120, 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]

(231−79, 231, 333)-Net over F₃ — Digital

Digital (152, 231, 333)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−79, 231, 4975)-Net in Base 3 — Upper bound on s

There is no (152, 231, 4976)-net in base 3, because

1 times m-reduction [i] would yield (152, 230, 4976)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 54 833674 579541 409187 078265 249205 997704 403301 460351 022406 188732 956329 417787 450539 102557 012810 655299 405565 795649 > 3²³⁰ [i]