MinT - Best Known (224−73, 224, s)-Nets in Base 3

Best Known (224−73, 224, s)-Nets in Base 3

(224−73, 224, 164)-Net over F₃ — Constructive and digital

Digital (151, 224, 164)-net over F₃, using

3¹ times duplication [i] based on digital (150, 223, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 43, 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 (107, 180, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 90, 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]

(224−73, 224, 377)-Net over F₃ — Digital

Digital (151, 224, 377)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−73, 224, 6409)-Net in Base 3 — Upper bound on s

There is no (151, 224, 6410)-net in base 3, because

1 times m-reduction [i] would yield (151, 223, 6410)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25094 887172 229642 095046 457706 246937 010117 308460 612258 044511 055229 534265 595199 817900 427894 044137 737285 619945 > 3²²³ [i]