MinT - Best Known (157−81, 157, s)-Nets in Base 3

Best Known (157−81, 157, s)-Nets in Base 3

(157−81, 157, 60)-Net over F₃ — Constructive and digital

Digital (76, 157, 60)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 55, 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, 102, 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]

(157−81, 157, 84)-Net over F₃ — Digital

Digital (76, 157, 84)-net over F₃, using

t-expansion [i] based on digital (71, 157, 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]

(157−81, 157, 533)-Net in Base 3 — Upper bound on s

There is no (76, 157, 534)-net in base 3, because

1 times m-reduction [i] would yield (76, 156, 534)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 273 943349 926134 876317 167406 612650 389272 364090 674296 476989 058169 671942 035985 > 3¹⁵⁶ [i]