MinT - Best Known (186−81, 186, s)-Nets in Base 3

Best Known (186−81, 186, s)-Nets in Base 3

(186−81, 186, 80)-Net over F₃ — Constructive and digital

Digital (105, 186, 80)-net over F₃, using

8 times m-reduction [i] based on digital (105, 194, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 97, 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]

(186−81, 186, 138)-Net over F₃ — Digital

Digital (105, 186, 138)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(186−81, 186, 1229)-Net in Base 3 — Upper bound on s

There is no (105, 186, 1230)-net in base 3, because

1 times m-reduction [i] would yield (105, 185, 1230)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18533 486847 826329 374943 115569 910231 382744 010263 759377 248602 547376 929292 022106 425272 266897 > 3¹⁸⁵ [i]