MinT - Best Known (159−86, 159, s)-Nets in Base 3

Best Known (159−86, 159, s)-Nets in Base 3

(159−86, 159, 56)-Net over F₃ — Constructive and digital

Digital (73, 159, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 58, 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 (15, 101, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(159−86, 159, 84)-Net over F₃ — Digital

Digital (73, 159, 84)-net over F₃, using

t-expansion [i] based on digital (71, 159, 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]

(159−86, 159, 449)-Net in Base 3 — Upper bound on s

There is no (73, 159, 450)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7602 735024 230665 868435 748698 675398 602834 739092 893325 490642 268512 074336 515137 > 3¹⁵⁹ [i]