MinT - Best Known (84, 84+89, s)-Nets in Base 3

Best Known (84, 84+89, s)-Nets in Base 3

(84, 84+89, 61)-Net over F₃ — Constructive and digital

Digital (84, 173, 61)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 57, 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, 116, 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]

(84, 84+89, 84)-Net over F₃ — Digital

Digital (84, 173, 84)-net over F₃, using

t-expansion [i] based on digital (71, 173, 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]

(84, 84+89, 590)-Net in Base 3 — Upper bound on s

There is no (84, 173, 591)-net in base 3, because

1 times m-reduction [i] would yield (84, 172, 591)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12345 289710 043498 853572 780173 048513 884823 604087 256610 957287 447640 725504 734779 700905 > 3¹⁷² [i]