MinT - Best Known (212−69, 212, s)-Nets in Base 3

Best Known (212−69, 212, s)-Nets in Base 3

(212−69, 212, 162)-Net over F₃ — Constructive and digital

Digital (143, 212, 162)-net over F₃, using

10 times m-reduction [i] based on digital (143, 222, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 111, 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]

(212−69, 212, 361)-Net over F₃ — Digital

Digital (143, 212, 361)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−69, 212, 6152)-Net in Base 3 — Upper bound on s

There is no (143, 212, 6153)-net in base 3, because

1 times m-reduction [i] would yield (143, 211, 6153)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47185 166565 867176 380286 179003 378966 685574 164675 531839 801345 123138 401137 841012 076884 342500 736397 977945 > 3²¹¹ [i]