MinT - Best Known (180−93, 180, s)-Nets in Base 3

Best Known (180−93, 180, s)-Nets in Base 3

(180−93, 180, 64)-Net over F₃ — Constructive and digital

Digital (87, 180, 64)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 61, 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]
2. digital (26, 119, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]

(180−93, 180, 84)-Net over F₃ — Digital

Digital (87, 180, 84)-net over F₃, using

t-expansion [i] based on digital (71, 180, 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]

(180−93, 180, 602)-Net in Base 3 — Upper bound on s

There is no (87, 180, 603)-net in base 3, because

1 times m-reduction [i] would yield (87, 179, 603)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 26 004874 663674 013341 294755 749169 464201 701375 220492 466984 351640 891173 699458 157326 675197 > 3¹⁷⁹ [i]