MinT - Best Known (143−69, 143, s)-Nets in Base 3

Best Known (143−69, 143, s)-Nets in Base 3

(143−69, 143, 61)-Net over F₃ — Constructive and digital

Digital (74, 143, 61)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 47, 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 (27, 96, 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]

(143−69, 143, 84)-Net over F₃ — Digital

Digital (74, 143, 84)-net over F₃, using

t-expansion [i] based on digital (71, 143, 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]

(143−69, 143, 632)-Net in Base 3 — Upper bound on s

There is no (74, 143, 633)-net in base 3, because

1 times m-reduction [i] would yield (74, 142, 633)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 57 395641 698464 194642 591131 169094 860376 952751 331009 100377 578109 943225 > 3¹⁴² [i]