MinT - Best Known (227−107, 227, s)-Nets in Base 3

Best Known (227−107, 227, s)-Nets in Base 3

(227−107, 227, 80)-Net over F₃ — Constructive and digital

Digital (120, 227, 80)-net over F₃, using

1 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]

(227−107, 227, 132)-Net over F₃ — Digital

Digital (120, 227, 132)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−107, 227, 1063)-Net in Base 3 — Upper bound on s

There is no (120, 227, 1064)-net in base 3, because

1 times m-reduction [i] would yield (120, 226, 1064)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 690974 889341 921816 795224 011335 157768 918946 844701 540859 698547 969839 216313 059120 438440 425442 015051 043682 838225 > 3²²⁶ [i]