MinT - Best Known (179−77, 179, s)-Nets in Base 3

Best Known (179−77, 179, s)-Nets in Base 3

(179−77, 179, 80)-Net over F₃ — Constructive and digital

Digital (102, 179, 80)-net over F₃, using

9 times m-reduction [i] based on digital (102, 188, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 94, 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]

(179−77, 179, 139)-Net over F₃ — Digital

Digital (102, 179, 139)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(179−77, 179, 1253)-Net in Base 3 — Upper bound on s

There is no (102, 179, 1254)-net in base 3, because

1 times m-reduction [i] would yield (102, 178, 1254)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8 639462 109299 212643 964554 122122 236406 860087 785632 940316 142610 655637 376737 172287 959125 > 3¹⁷⁸ [i]