MinT - Best Known (169−93, 169, s)-Nets in Base 3

Best Known (169−93, 169, s)-Nets in Base 3

(169−93, 169, 56)-Net over F₃ — Constructive and digital

Digital (76, 169, 56)-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 (15, 108, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(169−93, 169, 84)-Net over F₃ — Digital

Digital (76, 169, 84)-net over F₃, using

t-expansion [i] based on digital (71, 169, 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]

(169−93, 169, 453)-Net in Base 3 — Upper bound on s

There is no (76, 169, 454)-net in base 3, because

1 times m-reduction [i] would yield (76, 168, 454)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 149 027672 801568 661077 820378 509210 528913 456327 241869 193750 877427 010689 291850 394021 > 3¹⁶⁸ [i]