MinT - Best Known (163−85, 163, s)-Nets in Base 3

Best Known (163−85, 163, s)-Nets in Base 3

(163−85, 163, 60)-Net over F₃ — Constructive and digital

Digital (78, 163, 60)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 57, 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 (21, 106, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]

(163−85, 163, 84)-Net over F₃ — Digital

Digital (78, 163, 84)-net over F₃, using

t-expansion [i] based on digital (71, 163, 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]

(163−85, 163, 531)-Net in Base 3 — Upper bound on s

There is no (78, 163, 532)-net in base 3, because

1 times m-reduction [i] would yield (78, 162, 532)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 206793 327998 731979 653280 292886 023998 593768 895022 954405 259307 805672 402409 045785 > 3¹⁶² [i]