MinT - Best Known (124−57, 124, s)-Nets in Base 3

Best Known (124−57, 124, s)-Nets in Base 3

(124−57, 124, 64)-Net over F₃ — Constructive and digital

Digital (67, 124, 64)-net over F₃, using

trace code for nets [i] based on digital (5, 62, 32)-net over F₉, using
- net from sequence [i] based on digital (5, 31)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
    - a function field by GarcÃa and Quoos [i]

(124−57, 124, 85)-Net over F₃ — Digital

Digital (67, 124, 85)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(124−57, 124, 677)-Net in Base 3 — Upper bound on s

There is no (67, 124, 678)-net in base 3, because

1 times m-reduction [i] would yield (67, 123, 678)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 49550 640299 865092 010329 248258 496635 320214 044838 806531 631097 > 3¹²³ [i]