MinT - Best Known (127−53, 127, s)-Nets in Base 3

Best Known (127−53, 127, s)-Nets in Base 3

(127−53, 127, 80)-Net over F₃ — Constructive and digital

Digital (74, 127, 80)-net over F₃, using

5 times m-reduction [i] based on digital (74, 132, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 66, 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]

(127−53, 127, 113)-Net over F₃ — Digital

Digital (74, 127, 113)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−53, 127, 1057)-Net in Base 3 — Upper bound on s

There is no (74, 127, 1058)-net in base 3, because

1 times m-reduction [i] would yield (74, 126, 1058)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 330854 765505 046885 873682 211961 949162 114218 802729 796824 136389 > 3¹²⁶ [i]