MinT - Best Known (135−63, 135, s)-Nets in Base 3

Best Known (135−63, 135, s)-Nets in Base 3

(135−63, 135, 64)-Net over F₃ — Constructive and digital

Digital (72, 135, 64)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 46, 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 (26, 89, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]

(135−63, 135, 87)-Net over F₃ — Digital

Digital (72, 135, 87)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−63, 135, 686)-Net in Base 3 — Upper bound on s

There is no (72, 135, 687)-net in base 3, because

1 times m-reduction [i] would yield (72, 134, 687)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8658 564160 357639 601035 516670 301483 199167 762808 877963 434673 572995 > 3¹³⁴ [i]