MinT - Best Known (67, 67+78, s)-Nets in Base 3

Best Known (67, 67+78, s)-Nets in Base 3

(67, 67+78, 52)-Net over F₃ — Constructive and digital

Digital (67, 145, 52)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 52, 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 (15, 93, 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]

(67, 67+78, 72)-Net over F₃ — Digital

Digital (67, 145, 72)-net over F₃, using

net from sequence [i] based on digital (67, 71)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(67, 67+78, 420)-Net in Base 3 — Upper bound on s

There is no (67, 145, 421)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1629 582471 000596 544703 645081 099764 702471 600915 140195 336380 774585 407195 > 3¹⁴⁵ [i]