MinT - Best Known (217−103, 217, s)-Nets in Base 3

Best Known (217−103, 217, s)-Nets in Base 3

(217−103, 217, 76)-Net over F₃ — Constructive and digital

Digital (114, 217, 76)-net over F₃, using

5 times m-reduction [i] based on digital (114, 222, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 69, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 153, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(217−103, 217, 124)-Net over F₃ — Digital

Digital (114, 217, 124)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−103, 217, 991)-Net in Base 3 — Upper bound on s

There is no (114, 217, 992)-net in base 3, because

1 times m-reduction [i] would yield (114, 216, 992)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 551036 554043 892019 840431 419631 550916 690260 465925 528257 031086 806790 046164 977049 503228 160062 972415 449985 > 3²¹⁶ [i]