MinT - Best Known (36, 36+31, s)-Nets in Base 3

Best Known (36, 36+31, s)-Nets in Base 3

(36, 36+31, 42)-Net over F₃ — Constructive and digital

Digital (36, 67, 42)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (6, 21, 14)-net over F₃, using
  - net from sequence [i] based on digital (6, 13)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 6 and N(F) ≥ 14, 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]

(36, 36+31, 52)-Net over F₃ — Digital

Digital (36, 67, 52)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(36, 36+31, 389)-Net in Base 3 — Upper bound on s

There is no (36, 67, 390)-net in base 3, because

1 times m-reduction [i] would yield (36, 66, 390)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 31 570657 396731 678987 651701 702601 > 3⁶⁶ [i]