MinT - Best Known (156−85, 156, s)-Nets in Base 3

Best Known (156−85, 156, s)-Nets in Base 3

(156−85, 156, 52)-Net over F₃ — Constructive and digital

Digital (71, 156, 52)-net over F₃, using

1 times m-reduction [i] based on digital (71, 157, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 56, 24)-net over F₃, using
    - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (15, 101, 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]

(156−85, 156, 84)-Net over F₃ — Digital

Digital (71, 156, 84)-net over F₃, using

net from sequence [i] based on digital (71, 83)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(156−85, 156, 435)-Net in Base 3 — Upper bound on s

There is no (71, 156, 436)-net in base 3, because

1 times m-reduction [i] would yield (71, 155, 436)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 90 645406 755714 126077 858430 772621 589586 911900 837684 218889 257785 325345 780953 > 3¹⁵⁵ [i]