MinT - Best Known (69, 69+45, s)-Nets in Base 3

Best Known (69, 69+45, s)-Nets in Base 3

(69, 69+45, 80)-Net over F₃ — Constructive and digital

Digital (69, 114, 80)-net over F₃, using

8 times m-reduction [i] based on digital (69, 122, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 61, 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]

(69, 69+45, 121)-Net over F₃ — Digital

Digital (69, 114, 121)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+45, 1256)-Net in Base 3 — Upper bound on s

There is no (69, 114, 1257)-net in base 3, because

1 times m-reduction [i] would yield (69, 113, 1257)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 828303 112141 508465 814989 473818 849617 561034 717169 689897 > 3¹¹³ [i]