MinT - Best Known (216−105, 216, s)-Nets in Base 3

Best Known (216−105, 216, s)-Nets in Base 3

(216−105, 216, 75)-Net over F₃ — Constructive and digital

Digital (111, 216, 75)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 79, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
2. digital (32, 137, 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]

(216−105, 216, 116)-Net over F₃ — Digital

Digital (111, 216, 116)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−105, 216, 899)-Net in Base 3 — Upper bound on s

There is no (111, 216, 900)-net in base 3, because

1 times m-reduction [i] would yield (111, 215, 900)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 956675 054896 258072 907301 873449 978158 195577 147526 241959 806616 434490 405310 040859 453939 437340 447258 356289 > 3²¹⁵ [i]