MinT - Best Known (216−103, 216, s)-Nets in Base 3

Best Known (216−103, 216, s)-Nets in Base 3

(216−103, 216, 76)-Net over F₃ — Constructive and digital

Digital (113, 216, 76)-net over F₃, using

3 times m-reduction [i] based on digital (113, 219, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 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, 151, 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]

(216−103, 216, 122)-Net over F₃ — Digital

Digital (113, 216, 122)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−103, 216, 969)-Net in Base 3 — Upper bound on s

There is no (113, 216, 970)-net in base 3, because

1 times m-reduction [i] would yield (113, 215, 970)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 884140 623578 356489 692010 977680 203094 128993 322326 981233 044985 793957 736201 546353 588855 026025 330593 932497 > 3²¹⁵ [i]