MinT - Best Known (136, 136+111, s)-Nets in Base 3

Best Known (136, 136+111, s)-Nets in Base 3

(136, 136+111, 86)-Net over F₃ — Constructive and digital

Digital (136, 247, 86)-net over F₃, using

t-expansion [i] based on digital (134, 247, 86)-net over F₃, using
- 1 times m-reduction [i] based on digital (134, 248, 86)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (32, 89, 38)-net over F₃, using
      - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
        an explicitly constructive algebraic function field [i]
    2. digital (45, 159, 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]

(136, 136+111, 162)-Net over F₃ — Digital

Digital (136, 247, 162)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+111, 1398)-Net in Base 3 — Upper bound on s

There is no (136, 247, 1399)-net in base 3, because

1 times m-reduction [i] would yield (136, 246, 1399)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2355 388931 447020 818471 419825 888334 937530 527917 444900 293875 296642 569384 117662 533318 152020 348755 682976 751102 088643 116019 > 3²⁴⁶ [i]