MinT - Best Known (221−103, 221, s)-Nets in Base 3

Best Known (221−103, 221, s)-Nets in Base 3

(221−103, 221, 80)-Net over F₃ — Constructive and digital

Digital (118, 221, 80)-net over F₃, using

1 times m-reduction [i] based on digital (118, 222, 80)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 73, 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, 149, 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]

(221−103, 221, 133)-Net over F₃ — Digital

Digital (118, 221, 133)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−103, 221, 1085)-Net in Base 3 — Upper bound on s

There is no (118, 221, 1086)-net in base 3, because

1 times m-reduction [i] would yield (118, 220, 1086)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 950 825838 135134 458812 370473 419442 253778 248622 269923 064620 368852 102730 950610 012010 535145 718272 651287 920161 > 3²²⁰ [i]