MinT - Best Known (239−144, 239, s)-Nets in Base 3

Best Known (239−144, 239, s)-Nets in Base 3

(239−144, 239, 64)-Net over F₃ — Constructive and digital

Digital (95, 239, 64)-net over F₃, using

t-expansion [i] based on digital (89, 239, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(239−144, 239, 96)-Net over F₃ — Digital

Digital (95, 239, 96)-net over F₃, using

t-expansion [i] based on digital (89, 239, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(239−144, 239, 462)-Net in Base 3 — Upper bound on s

There is no (95, 239, 463)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 202108 307803 008927 235687 237211 581873 168297 857823 579825 180855 969989 104865 702587 005943 755214 917858 684824 155365 736561 > 3²³⁹ [i]