MinT - Best Known (225−109, 225, s)-Nets in Base 3

Best Known (225−109, 225, s)-Nets in Base 3

(225−109, 225, 76)-Net over F₃ — Constructive and digital

Digital (116, 225, 76)-net over F₃, using

3 times m-reduction [i] based on digital (116, 228, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 71, 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, 157, 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]

(225−109, 225, 121)-Net over F₃ — Digital

Digital (116, 225, 121)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(225−109, 225, 946)-Net in Base 3 — Upper bound on s

There is no (116, 225, 947)-net in base 3, because

1 times m-reduction [i] would yield (116, 224, 947)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 75303 600465 058546 038386 711997 006182 975930 454737 566517 443088 774761 225134 724209 676966 280236 747714 445066 917789 > 3²²⁴ [i]