MinT - Best Known (51, 51+35, s)-Nets in Base 3

Best Known (51, 51+35, s)-Nets in Base 3

(51, 51+35, 80)-Net over F₃ — Constructive and digital

Digital (51, 86, 80)-net over F₃, using

trace code for nets [i] based on digital (8, 43, 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]

(51, 51+35, 89)-Net over F₃ — Digital

Digital (51, 86, 89)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+35, 855)-Net in Base 3 — Upper bound on s

There is no (51, 86, 856)-net in base 3, because

1 times m-reduction [i] would yield (51, 85, 856)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36170 881801 001207 041214 919425 562792 314289 > 3⁸⁵ [i]