MinT - Best Known (225−105, 225, s)-Nets in Base 3

Best Known (225−105, 225, s)-Nets in Base 3

(225−105, 225, 80)-Net over F₃ — Constructive and digital

Digital (120, 225, 80)-net over F₃, using

3 times m-reduction [i] based on digital (120, 228, 80)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 75, 32)-net over F₃, using
    - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, 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]

(225−105, 225, 135)-Net over F₃ — Digital

Digital (120, 225, 135)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(225−105, 225, 1098)-Net in Base 3 — Upper bound on s

There is no (120, 225, 1099)-net in base 3, because

1 times m-reduction [i] would yield (120, 224, 1099)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 78446 806347 353177 864815 088430 468322 271285 420955 249485 130904 762506 592912 645579 491806 036475 655845 367768 302969 > 3²²⁴ [i]