MinT - Best Known (170−87, 170, s)-Nets in Base 3

Best Known (170−87, 170, s)-Nets in Base 3

(170−87, 170, 61)-Net over F₃ — Constructive and digital

Digital (83, 170, 61)-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 (27, 114, 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]

(170−87, 170, 84)-Net over F₃ — Digital

Digital (83, 170, 84)-net over F₃, using

t-expansion [i] based on digital (71, 170, 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]

(170−87, 170, 591)-Net in Base 3 — Upper bound on s

There is no (83, 170, 592)-net in base 3, because

1 times m-reduction [i] would yield (83, 169, 592)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 430 424521 727357 258765 388656 074232 919997 982612 244152 597863 615515 133888 431960 098625 > 3¹⁶⁹ [i]