MinT - Best Known (99, 99+81, s)-Nets in Base 3

Best Known (99, 99+81, s)-Nets in Base 3

(99, 99+81, 80)-Net over F₃ — Constructive and digital

Digital (99, 180, 80)-net over F₃, using

2 times m-reduction [i] based on digital (99, 182, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 91, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(99, 99+81, 123)-Net over F₃ — Digital

Digital (99, 180, 123)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 99+81, 1037)-Net in Base 3 — Upper bound on s

There is no (99, 180, 1038)-net in base 3, because

1 times m-reduction [i] would yield (99, 179, 1038)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 26 102778 315788 602622 442061 927160 298870 217081 206189 172158 552627 056088 175580 970220 958865 > 3¹⁷⁹ [i]