MinT - Best Known (211−68, 211, s)-Nets in Base 3

Best Known (211−68, 211, s)-Nets in Base 3

(211−68, 211, 164)-Net over F₃ — Constructive and digital

Digital (143, 211, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 41, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (102, 170, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 85, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(211−68, 211, 371)-Net over F₃ — Digital

Digital (143, 211, 371)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−68, 211, 6152)-Net in Base 3 — Upper bound on s

There is no (143, 211, 6153)-net in base 3, because

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]