MinT - Best Known (34, 65, s)-Nets in Base 3

Best Known (34, 65, s)-Nets in Base 3

(34, 65, 40)-Net over F₃ — Constructive and digital

Digital (34, 65, 40)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 19, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. 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]

(34, 65, 47)-Net over F₃ — Digital

Digital (34, 65, 47)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 65, 334)-Net in Base 3 — Upper bound on s

There is no (34, 65, 335)-net in base 3, because

1 times m-reduction [i] would yield (34, 64, 335)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 516058 054742 880350 073928 473891 > 3⁶⁴ [i]