MinT - Best Known (135, 135+111, s)-Nets in Base 3

Best Known (135, 135+111, s)-Nets in Base 3

(135, 135+111, 86)-Net over F₃ — Constructive and digital

Digital (135, 246, 86)-net over F₃, using

t-expansion [i] based on digital (134, 246, 86)-net over F₃, using
- 2 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]

(135, 135+111, 160)-Net over F₃ — Digital

Digital (135, 246, 160)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+111, 1370)-Net in Base 3 — Upper bound on s

There is no (135, 246, 1371)-net in base 3, because

1 times m-reduction [i] would yield (135, 245, 1371)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 807 036852 556582 276762 816756 065419 663272 362505 307444 699762 350725 549992 197101 333745 951335 017689 666015 008195 205401 263491 > 3²⁴⁵ [i]